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on what might be up for discussion as the two leaders tuck into their burgers at the Downing Street barbecue later.
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We're expecting President Obama to arrive at Downing Street shortly - he'll be speaking to Deputy PM Nick Clegg too while he is there.
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From the BBC's Will Walden: Noticeable ramping up of security around the Parliamentary estate this morning ahead of the President's visit. Sniffer dogs, bag searches on pass holders and visitors alike, and hand held body scanners before you can pass go. First time I've ever had to queue at security here.
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UK Foreign Secretary William Hague has been on the BBC's Today programme - rejecting suggestions that the US was taking a "back seat" in the campaign in Libya. Mr Hague told Today: "We couldn't have done what we did at the beginning of the military operations in Libya - the destruction of the Gaddafi regime's fixed air defences - without the unique assets of the United States."
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The BBC's Laura Kuenssberg says Downing Street is now packed with press photographers hoping to get a snap of the US President. The red carpet has been laid out in preparation for his arrival.
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At Buckingham Palace President Obama's "outriders" are circling while they wait for the gates to open, before the president's car leaves for Downing Street
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President Obama's convoy is on its way down the Mall en route to Downing Street.
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The president's car is now heading down Whitehall - where the traffic has been stopped - and will be arriving at Downing Street any minute.
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David Cameron gets out to welcome the president, they shake hands and pose for the obligatory pictures on the doorstep of No 10.
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The two leaders are followed in by President Obama's entourage - and the White House press corps.
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From BBC Political Correspondent Ross Hawkins: No detail of the barbecue to be hosted by Michelle Obama and Samantha Cameron is being left to chance. Civil servants in the Cabinet Office have just received a staff message warning those who might not have heard about the event not to worry if they see smoke rising in the Downing Street area today.
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A quick recap: President Obama and David Cameron are now inside Downing Street for talks - but will resurface for a barbecue at Number 10 for military personnel later in the morning. Later on the President will be giving a speech on US foreign policy to MPs and peers in Parliament before returning to Buckingham Palace.
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President Obama's speech in Parliament later will apparently aim to be upbeat and promise a "new dawn" for the West after difficult decade. Hillary Clinton and William Hague are expected to arrive at Downing Street from the Foreign Office any minute now.
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Labour's Douglas Alexander was at his party leader Ed Miliband's meeting with President Obama on Tuesday. He says he is a "president at the top of his game" and is "impressive" to meet in person. The Democrats are "our crowd in the United States," says Mr Alexander but he is, nevertheless, pleased that the British prime minister has a good working relationship with President Obama.
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William Hague and Hillary Clinton have arrived at Downing Street - Mrs Clinton waved as she walked past but chose to shrug off photographers' requests for her to pose on the doorstep for a picture. She got a few boos for that from the waiting snappers.
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A bit of news from inside No 10 from the Press Association. As Mr Cameron and the US President sat down for talks in the White Room at Downing Street, they were surrounded by photographers who started snapping away. The president joked: "All right guys, one of those must have worked" as they tried to get down to work.
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How are the US media covering President Obama's visit? The
New York Times looks at the presents the Obamas exchanged with the Royals and some of the details of their visit to Buckingham Palace.
USA Today says the couple were fawned over and treated like royalty on Tuesday.
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Here's a rare shot from inside the Downing Street flat of First Lady Michelle Obama and Samantha Cameron - courtesy of the White House. For those thinking that those outfits look familiar, the picture was actually taken on Tuesday. The two women are teaming up to host a barbecue for military personnel in the No 10 garden shortly. At least the curse of the British barbecue - the rain - is nowhere to be seen.
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writes: It's amazing to see the close relationship between the UK and US grow and flourish. America's relationship with the UK is by far the most critical and important bilateral relationship for both nations because we can always rely on each other for support.
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From BBC Political Correspondent Ross Hawkins: Remarkable shots of the Cabinet room in Downing St give the impression David Cameron has made some rather high profile appointments to his government. The President sits alongside Hillary Clinton at the famous coffin-shaped table, opposite the prime minister, political advisors and ministers fill the other seats.
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The BBC's Laura Kuenssberg said: President Obama's driver has been using the break, while he's in Downing Street, to polish the bonnet of the presidential car.
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From BBC Political Correspondent Ross Hawkins: Pictures from inside Downing St appear to show Nick Clegg joking with President Obama about how the President handled the awkward moment when the national anthem played while he was still proposing a toast at last night's banquet. "I thought you did exactly the right thing," Mr Clegg reassures him.
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Back to that "awkward moment" at last night's banquet. President Obama joked to Mr Clegg that he thought the moment when the orchestra struck up, as he started speaking, was like something "out of the movies" and a "soundtrack".
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President Obama needn't worry about his return trip - apparently the threat from the volcanic ash cloud, that caused him to cut short his trip to Ireland - is receding across Europe, although it's still causing problems in Germany.
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It's now the First Lady's turn to leave Buckingham Palace for Downing Street - she has a pretty impressive motorcade too, which is currently heading down to Downing Street.
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Here's a quick round-up of some of the coverage in the British press. President Obama gets a big thumbs up in The Mirror, which describes him as the "king of cool" while the Daily Mail notices that he seems to have got the date wrong when he signed his name in the Westminster Abbey guest book. Meanwhile the Guardian's cartoonist Steve Bell has his own take on the visit while the Telegraph praises the First Lady's fashion sense
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Michelle Obama has managed to bypass the waiting photographers and cameramen to enter Downing Street via the back door. She's getting ready for the barbecue for service personnel.
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Here's that picture from earlier, as President Obama and Hillary Clinton took part in a cabinet meeting. Looks like they're having fun...
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From the BBC's Chris Mason: Hillary Clinton and William Hague leave 10 Downing St and return to the Foreign Office. Home Secretary Theresa May also just left.
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We're expecting the US President and David Cameron to take journalists' questions from about 1230 BST - expect them to be asked about Libya, the "special relationship" and the economy. After his speech in Parliament later, it will be Mr Obama's turn to play host to the Queen. She and Prince Philip will be his guests at the US Ambassador's residence in Regents Park for the last night of the state visit.
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Now the news you've all been waiting for. After the grandeur of last night's state banquet at Buckingham Palace, we are told the Downing Street barbecue is a little more down to earth. Guests are apparently tucking into British sausages, beefburgers, Kentish lamb chops, corn on the cob, Jersey Royal potatoes, with tomato, mozarella and basil salad, then summer berries and ice cream to top it off. Sounds tasty.
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Here's some of the pictures of the day so far, which we'll be adding to throughout Wednesday
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President Obama is set to put the past behind him when he addresses Parliament, writes the BBC's Washington editor Mark Mardell.
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Looks like David Cameron and Barack Obama have decided to ditch the jackets for the barbecue - although with all the military uniforms around it's still a bit more formal than the average grill-up. The two leaders have been shaking hands and chatting to service personnel in the garden, as well as dishing out the burgers. Samantha Cameron and Michelle Obama have been serving up the side dishes.
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It's all smiles at the barbecue as the two leaders relax a bit following formal talks at Downing Street. Wonder what the military personnel present think about being served burgers by some of the most powerful politicians in the world?
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From the BBC's Ross Hawkins: The president and prime minister took a bit of advice from the catering staff before, sleeves rolled up, attempting to flip and serve burgers themselves. Both are careful not to get barbecue grease on their ties.
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President Obama and David Cameron are due to give a press conference shortly and are expected to present a united front. But there have been some tensions over the scale of US involvement in action in Libya - Vice chairman of the Commons defence committee, Tory MP Mark Pritchard, tells us: "The United States military is doing much; but the reluctance of President Obama, for primarily domestic political reasons, not to fully commit the United States to the conflict in Libya, could unhelpfully prolong the conflict".
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Former Lib Dem leader Sir Menzies Campbell says President Obama is very comfortable with this sort of occasion and the two leaders appear to have a "natural affinity".
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The jackets are back on and President Obama and David Cameron have left the barbecue for the short journey to Lancaster House - where they will be taking journalists' questions.
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The traffic on Whitehall has been stopped as "The Beast" - the president's imposing armoured car - makes its way to Lancaster House.
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Will the barbecue bonhomie continue as the two leaders face questions from the world's media? It was all smiles as they arrived at Lancaster House, and the press conference is taking place outside in the sunshine, which always helps the mood.
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writes: I think it's great to see Barack Obama and his wife Michelle visiting our country. But I am embarrassed by the media coverage in the newspapers and the so called fashion experts comparing the dresses worn by the Duchess of Cambridge and the First Lady. Is this a point scoring exercise to gauge the value of USA's presidential visit? Comments from these so called fashion experts should be barred by super-injunctions. We need to gag the lot of them!
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Ahead of the president's address to Parliament later, former Commons Speaker Baroness Boothroyd has been remembering Nelson Mandela's speech in Westminster Hall in 1996. She told the BBC World Service it was the "highlight of my Speakership": "I'd warned him to be careful of the steps of Westminster Hall, worn by 1,000 years of history. When we met at St Stephens entrance I said: 'Now Mr President, you pace it and I will take my time with you, but I'm concerned about those steps for you'. He turned to me and said: 'Don't worry Madam Speaker, I came here at 6 o clock this morning to pace them for myself'. With that the trumpets sounded, he took my hand and we walked into Westminster Hall. It really was magnificent."
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The press corps are settled in, the flags are flying and the sun's out - but no sign as yet of President Obama and David Cameron. Perhaps they're comparing notes on Libya and Afghanistan - two topics they are expected to be grilled on.
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Might the case of Gary McKinnon - the computer hacker facing extradition to the US - be raised at the press conference? BBC diplomatic affairs correspondent James Robbins thinks it's the sort of thing that will have been mentioned during private talks between the two leaders.
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The BBC's James Robbins also thinks that there might also be questions about funding for Egypt and Tunisia, as they emerge from the "Arab Spring".
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The two leaders are heading out for the press conference
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David Cameron kicks off the press conference by talking about the barbecue and the debt everyone owes the military. He also jokes it was probably the first time a UK PM had given a US president "a bit of a grilling".
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Mr Cameron goes on to praise President Obama and mentions the tornado in Missouri. He calls the president by his first name as he talks about joint UK-US ventures and the "special" relationship, which he says is a "living, working partnership" essential to security and prosperity.
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Mr Cameron says the talks this morning have been about creating jobs and keeping "our people safe". He again refers to "Barack and I" as he talks about the need to address national debt and invest in railways, science, innovation and young people
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Mr Cameron says there were also good discussions on security and terrorism, saying of terrorism that both countries had "suffered at its hands". He mentions that his wife was in New York on 9/11 and talks of the sympathy felt for those who lost their loved ones.
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The UK PM congratulates Mr Obama for the Bin Laden operation which he says struck "at the heart" of international terrorism. He moves on to Pakistan, and says they will not "walk away" but will work more closely with Pakistan.
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David Cameron also says Afghanistan will be a high priority and congratulates President Obama for his "bold" speech on the Middle East peace process. The road will be "long and arduous" but the prize is clear, he says - security and weakening the cause of extremists.
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Mr Cameron then moves on to the uprisings in the Arab world, saying it is not a time for the West to "shrink back" as it is "massively in our interests" to support democracy, to make the world a safer place. He says they will be pushing for a major economic support programme for countries seeking reform.
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Col Gaddafi "must go" in Libya, Mr Cameron says. The PM says democracy is "built from the ground up" and "real change" takes time.
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After a joke about the weather, Mr Obama says on a whole host of issues he and Mr Cameron "see eye to eye" despite being from different political traditions.
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The President says the "special relationship" is "stronger than it has ever been" and he wants it to stay that way.
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The BBC's Ross Hawkins says: This bit of international diplomacy is already looking ahead to the next - at the G8. David Cameron promises to push for a major programme of economic and political support for countries seeking reform.
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President Obama moves on to economic growth and international security and says they are close to "turning a corner" in Afghanistan - stressing the need for a political settlement there.
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The US is committed to a "strong enduring partnership" with the people of Afghanistan, President Obama says. He adds that the US will oppose efforts to silence those in Arab countries who are campaigning for freedom - and goes on to cover Libya, Syria and Yemen in his remarks.
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President Obama thanks David Cameron for his supportive comments about his Middle East peace process speech and mentions that joint initiative to support military families, following the "outstanding barbecue" at No 10.
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David Cameron says the President and he both believe they should be "turning up the pressure" in Libya, within the terms of the UN resolution. In response to a question about how different their relationship is, to that of George Bush and Tony Blair, he adds that every relationship between a UK PM and a US president is different: "We have to learn the lessons of history about how best we promote the values that we share", including a "patient understanding that building democracy takes time".
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President Obama says the action in Libya has "saved lives" and was done as part of a broad coalition, under a UN mandate. He says there will "not be a let up in the pressure we are applying". The goal is to make sure that the Libyan people can decide how to proceed, he says.
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The BBC's Ross Hawkins says: As the leaders speak, Number 10 releases a list of six US / UK joint projects including a joint task force to help service personnel and veterans and a "UK-US Joint Strategy Board" which it says will help to develop a co-ordinated approach to long term security and economic challenges.
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On Afghanistan President Obama stresses there is a "broad based international mandate" for the action there and says he is confident they can achieve a political solution there. There is "no doubt" that the US and UK have a special relationship, he says: "There is so much that binds us together, it's not surprising that on the international stage we are going to be working together, rather than at cross purposes."
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A question from a US journalist on Col Gaddafi - will the US be committed to the campaign until he is removed from power? President Obama says his goal is "to protect people on the ground" and give Libyans the space to bring about a change towards democracy. He says the US "moved very heavily on the front end" - with initial air strikes - but says it will be "difficult" to meet the UN mandate of security for the Libyan people, while Gaddafi is attacking them. "At minimum" he says they will make sure Gaddafi cannot "send in a bunch of thugs" to kill innocent civilians.
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President Obama says "David and I both agree" they cannot put "boots on the ground" in Libya - saying that limits the air operations and it is up to the opposition on the ground as well. There are no "secret" air assets that can quickly sort out the situation in Libya, he adds.
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Amid suggestions the US is taking a back foot on Libya, David Cameron says a "huge number" of air sorties on Libya come from the US, which has "unique assets and capabilities" that others do not have.
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On UK computer hacker Gary McKinnon, facing extradition to the US, President Obama says the case is now in the hands of the British legal system and they will be "respectful of that process".
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The BBC's Ross Hawkins says: The President pours cold water on any hope of a sudden, devastating US intervention in Libya. There may be a "false perception" that the US has a "secret super effective air assets that are in a warehouse somewhere", he says. He promises "slow steady" progress, but some in London might want more.
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President Obama says countries have to deal with their deficits differently and at a different pace, but they all want to "arrive at the same point", achieving prosperity in a "responsible" way.
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President Obama adds that countries have to take a "balanced approach" - a mix of cuts but also thinking about how to generate revenues - to deal with their deficits.
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David Cameron says he understands concern about Gary McKinnon - saying the case is currently before the home secretary.
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On deficit reduction, Mr Cameron says he agrees each country has different circumstances. Unlike the US, Britain does not have a reserve currency, he adds.
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The US and UK share similar values about not loading debt onto future generations, adds the British prime miniser, and "taking responsibility" to deal with their own deficits.
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The two leaders have made carefully nuanced efforts to sound in sympathy on economic policy - but David Cameron's opponents will no doubt leap on anything that sounds like President Obama questioning the way the UK government intends to cut the deficit, says the BBC's Ross Hawkins.
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President Obama is asked again about the Middle East peace process. He says his goal is a Jewish state of Israel that is safe and secure and recognised by its neighbours and a sovereign state of Palestine: "I'm confident that can be achieved" he says, but it will require "wrenching compromise" by both sides.
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This has been a very serious press conference - the jokes and smiles of the table tennis game and the barbecue have been left far behind as the two leaders seek to address international issues and stress the closeness of their relationship.
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President Obama says he recognises Israel's concern about Hamas and accepts it is "very difficult" to sit down to negotiate with a party that denies your right to exist - he says he does not want the Palestinians to forget they have obligations as well, if there is to be any prospect of peace "moving forward".
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David Cameron says the US and UK will always stand up for Israel's right to exist - but the Palestinians need to know the two countries understand their need for dignity and a Palestinian state. Neither side now has the "excuse" to stand aside from talks, he says.
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When you look at what had to happen in Northern Ireland for "peace to come about", Mr Cameron says, there had to be some recognition of what was needed on both sides. President Obama said it was "inspiring to see" people in Northern Ireland changing how they thought about themselves and those who were once considered enemies. He said it was an "enormous source of hope" and showed if you "stick to it" ultimately even the worst of conflicts could be resolved.
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And with that, the joint press conference ends. It was dominated by questions about international events. President Obama in his closing remarks thanked Mr Cameron for mentioning the storms in Missouri, saying the US has been "battered" in recent months and it was good to know they were in their allies' thoughts and prayers
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BBC North America editor Mark Mardell says he saw a strong difference in approach between the two leaders on economic matters. Mr Obama admitted that sometimes governments might try things to generate growth but they don't always work - and in that case, they should move on and try something else. Mardell says he can't imagine that sort of frankness from David Cameron.
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Looking ahead now, the leaders are off for a private lunch and a bit of down-time before Mr Obama's big speech at Westminster Hall. That was due to take place at about 3.40pm but timing could slip a bit. Mrs Obama, meanwhile, is heading to Oxford University for an open day with some north London school children.
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BBC News education correspondent Sean Coughlan says: Inside the historic dining hall at Christ Church at Oxford University, where Michelle Obama is going to talk to some inner-city London school pupils, it looks like a scene from Hogwarts. Mind you, that shouldn't be surprising because this college is where some of the Harry Potter movies were filmed.
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Michael White, from the Guardian, says the press conference was "smooth" and full of "warm words". But like Mark Mardell, he also picked up on the divergence on matters financial, noting that David Cameron "did not get an endorsement of the coalition's economic strategy, sharp cuts upfront to ensure renewed growth".
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BBC News education correspondent Sean Coughlan, in Oxford, says: One for the security and dog buffs. The sniffer dog checking visitors to the Michelle Obama event is a "sprocker". A springer spaniel mixed with a cocker. Who said you can't teach a new dog old tricks?
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A bit more on those joint US-UK projects. We understand they include working more closely together on higher education - offering more post-graduate opportunities in institutions on both sides of the Atlantic - and on tackling cyber crime. There will also be more co-operation on international development, with closer ties between the US Peace Corps and the UK charity VSO.
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Rosa Prince, from the Daily Telegraph, found the comparison between Northern Ireland and Israel/Palestine at the press conference interesting. David Cameron said that in the latter situation, "both sides needed to gain some undertanding of the position of the other before peace could be contemplated", she says. And she adds: "He implied, but didn't say, that that mental shift has yet to take place in the former."
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Rory Stewart MP, a Conservative member of the Foreign Affairs Select Committee, says that while it's nice that David Cameron and Barack Obama get on well, it would be a terrible state of affairs if something as important as US-UK relations boiled down to personalities.
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At first it's hard to hear Mrs Obama over the clatter of flash bulbs from the massed ranks of photographers, but she's given a microphone to give her a helping hand.
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Over to now and Mrs Obama says she's "beyond thrilled" to be at Christ Church College, Oxford, as part of an open day for school children.
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The First Lady says her visit to Elizabeth Garret Anderson School two years ago was her first solo trip as First Lady. She says she was inspired by the pupils passion and energy and felt a strong sense of connection with them. "In your stories I saw so much of my story," she says.
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"All of us believe that you belong here," she tells the pupils. The aim of her visit is to encourage girls from less than privileged backgrounds to apply to top universities. She says when she was young, people questioned whether someone with her upbringing could succeed and she worried whether they might be right. "But after a few months, I realised I was just as capable and I had just as much to offer," she adds.
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Away from Oxford, it seems like that difference in economic approach between Mr Cameron and Mr Obama is becoming a bit of a talking point. The New Statesman's George Eaton says "the Tories were desperate" for an endorsement from the president for their deficit reduction strategy, but "he didn't even come close".
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writes: Whilst I heartily welcome the affirmation of the 'special relationship' and would be happy for far closer ties with the US in place of the strained and false relationship with Europe I bear in mind that the relationship only works when it is to the mutual benefit of both sides and will slip easily away should those conditions or the political landscape change. Might be nice of they could negotiate a relaxed visa regime so that Brits can live and work in the US without having to go through the rigmarole of the present visa application system.
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Peter Hoskin, from the Spectator, says the economic differences show the leaders' "mutual appreciation has its limits". But he goes on: "If Obama was being completely upfront, he might have admitted that his plan is actually to cut faster and deeper than Osborne... As it was, it suited the President to sound more restrained than that today."
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In a question and answer session in Oxford, Michelle Obama is asked how long she thinks it'll be before there's a female US president. "I think we're there," she replies. "Our job as women is to envision ourselves as leaders and be ready to battle." She describes Hillary Clinton as "a fabulous leader" and says she was an "outstanding" competitor in the fight for the White House.
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writes: Its nice they're having a bbq. I can't afford meat anymore. Well done prime minister - or is that prime steak?
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Michelle Obama took questions from the girls Here's Michelle Obama talking to those pupils in Oxford.
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When you first met your husband, did you think he'd go on to do such great things?, Mrs Obama is asked by one of the pupils. "Absolutely not" she says - then adds: "Just kidding." "I knew he was special," she says, "He was smart, he worked hard, he was low-key, he was funny, he was cute... But I didn't think he was going to be president until the night we won."
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Back in London, Westminster Hall is filling up. It really is a spectacular setting for the speech. When he steps up today, Barack Obama will become only the fourth head of state - and the first US President - to address both Houses of Parliament in the hall.
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Front and centre in Westminster Hall are former Labour prime ministers Tony Blair and Gordon Brown, chatting away to each other. It's been a while since we've seen that.
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We're now watching President Obama - in his official car, the Beast - leave Buckingham Palace for Parliament.
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There's a sizeable crowd awaiting him outside the Houses of Parliament - a mixture of dedicated fans and curious tourists. Not much sign of any protests.
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Here's a snapshot of Westminster Hall, which is packed with politicians awaiting President Obama's speech
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Sir Christopher Meyer, former UK ambassador to the US, tells the BBC that no-one in either the Obama or Cameron camps will want this visit to be just about pomp and ceremony. He says the message will be that serious things have also been discussed and decided.
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David Cameron has taken his place inside Westminster Hall.
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President Obama's car has now arrived at the Palace of Westminster. A trumpet fanfare sounds as he steps out.
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Mr Obama is greeted by Black Rod and the Lord Chamberlain. He then meets the Speakers of the Commons and the Lords before walking inside.
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President Obama goes for a private tour of the building.
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Here's a quartet you don't see every day. Brown, Blair, Major and Clegg - if only we could lip read.
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The audience in Westminster Hall is being treated to some music while they wait. There is polite applause as one particular number is completed.
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BBC Washington correspondent Katty Kay says it's no surprise that Mr Obama asked for a private tour of the Palace of Westminster because history, sentimentality, tradition is so important to the US-UK bond.
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And with a trumpet fanfare, President Obama enters Westminster Hall.
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The president takes his seat on the stage. He's introduced to the assembled masses by Commons Speaker John Bercow.
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"Few places reach so far into the heart of our nation," says Mr Bercow, of the hall itself. He welcomes the President "as our friend and a statesman" - and, for the US audience, quotes Abraham Lincoln.
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Barack Obama now gets to his feet. He begins by saying he has known "few greater honors than the opportunity to address the Mother of Parliaments at Westminster Hall".
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"I'm told the last three speakers here have been the Pope, Her Majesty the Queen, and Nelson Mandela, which is either a very high bar or the beginning of a very funny joke," the president says - cue laughter all round.
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BBC Deputy Political Editor James Landale says: The President notes Britain and the US have had their ups and downs over history, pointing to "a small scrape about tea and taxes". A joke maybe but remember that in his inaugural speech as President his first reference to Britain was as "the enemy" as he told a story about the war of independence.
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BBC Deputy Political Editor James Landale says: The President has addressed the special relationship question head on - noting that it is often overanalysed for signs of "strains and stress". His point is that the relationship works not because of shared history but shared values. This is the pragmatic note that he and David Cameron are trying to emphasise.
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The beginnings of democracy on this island inspired millions around the world - but nowhere more so than among "the rebel-rousing colonists" across the Atlantic, Mr Obama says.
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BBC Deputy Political Editor James Landale says: The President is now taking a swift walk through history to illustrate the shared values of the US and the UK - from the Magna Carta's influence on the Declaration of Independence through to the Normandy landings.
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"We are the allies who landed at Omaha and Gold who sacrificed side by side to free a continent from the march of tyranny," says the president. I wonder if he's doing all this from memory - he certainly isn't looking down at any notes.
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The president says the UK and US are at "a pivotal moment" and "profound challenges" lie before both. He goes on to talk about terrorism, the global economy and the uprisings in the Middle East and north Africa.
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BBC Deputy Politcal Editor James Landale says: The President's first substantive point. The world is at a "pivotal moment". There's been a "diffficult decade" but troops are leaving Iraq, Afghanistan is in transition and the fight against al Qaeda has been dealt a huge blow. But there is now a new international order. And it is one with new challenges that still requires leadership from the "indispensible" alliance between the US and the UK. We remain, he says, "the greatest catalyst for global action".
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Mr Obama says the rise of new nations to strength does not mean that the US and UK no longer matter. In fact, he says: "The time for our leadership is now." The camera cuts to Foreign Secretary William Hague who nods his head sagely in agreement.
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The President in full flow...
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Mr Obama is talking about matters economic and the power of the free market. He says that in order to stay ahead in an increasingly competitive world, countries like the UK and US will have to "redouble our investments in science and engineering, and renew our national commitments to educating our workforces".
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Something on that economic divergence between the UK and US that we mentioned earlier. Mr Obama says the both countries must take care of its citizens while "ensuring that we're not consumed with a level of debt that could sap the strength and vitality from our economies". He adds: "That will require difficult choices and different paths for both of our countries." Note the word "different"...
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"Our nations are not and never will be at war with Islam," the president says, while talking about terrorism. A sober-faced David Cameron nods his head at that.
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On nuclear weapons, Mr Obama says firmly that "from North Korea to Iran, we have sent a message that those who flaunt their obligations will face consequences".
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writes: Unexpectedly, Obama's entrance brought me to tears. He represents an America that is the hope of the whole world, way beyond economics and politics but about the importance of one individual. Obama's success inspires the one person anywhere who tries to chalenge oppression.
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BBC Deputy Political Editor James Landale says: The President said the UK and US are on the side of those struggling for democracy in the Middle East and North Africa. "Now we must show that we will back up these words with deeds. That means investing in the future of those nations that transition to democracy, starting with Tunisia and Egypt - by deepening ties of trade and commerce; by helping them demonstrate that freedom brings prosperity. And that means standing up for universal rights - by sanctioning those who pursue repression, strengthening civil society, and supporting the rights of minorities." What that means is more cash and trade and aid for Tunisia and Egypt - which many diplomats think the west has neglected since their regimes were overturned. What it means for Syria - which he did not mention by name - is some kind of undefined "sanction".
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"History tells us that democracy is not easy. Power rarely gives up without a fight. Given what was said earlier about Libya, the president must surely have Col Gaddafi in mind when he says that.
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As he moves from topic to topic, the words "shared interest", "common goal" and, above all, "we" are repeated again and again.
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BBC Deputy Political Editor James Landale says: The President is telling his critics that the US did not drag its feet over Libya, that it is engaged, and that massacre has been avoided. But equally, note the lack of mention of Gadaffi by name, just an oblique mention of the "shadow of tyranny".
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For the first time, the speech is interrupted in by applause. The spontaneous moment comes after Mr Obama says it is that acceptance of diversity - and of newcomers - that makes it possible "for the grandson of a Kenyan who served as a cook in the British Army to stand before you as President of the United States".
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Just been told the President doesn't, in fact, have a super-human memory - he has an autocue.
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And with that, he's done. Cue standing ovation.
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Lords Speaker Baroness Hayman thanks the President for his speech. She says his words about the future of the UK-US relationship - and what the two countries can achieve together - were particularly inspirational.
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"Great speech," says Nick Clegg, as he shakes the President's hand - Mr Obama also exchanges pleasantries with David Cameron, Gordon Brown, Sir John Major and Tony Blair. He then makes his way down the central aisle of Westminster Hall, shaking lots more hands as he goes.
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The applause continues, as President Obama makes his way through the adoring crowd, accompanied by John Bercow.
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writes: Not many true statesmen on the world stage. Obama straddles the world as a "giant"!
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writes: It's always stirring for most of us in America to see our president welcome in the place to which we trace our own democratic ideals (a couple of little 18th & 19th century wars notwithstanding). In our media, Obama is almost swamped by our own party strife...and that's probably just our messy democracy working well. Seeing his reception in London reminds me of the stunning progress his election represents for our country. I might not vote for him....but I'm proud of how well he represents us to our most important friend, the UK.
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Apparently the barbecue weather doesn't suit everyone. Andrew Porter, of the Daily Telegraph, reports that a member of the audience in Westminster Hall collapsed in their seat, and had to be wheelchaired out of the hall. He says it was no wonder given the building is "decidedly unairconditioned".
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Mr Obama is still on his marathon hand-shaking mission. He's chatting animatedly to MPs and peers, greeting politicians of all party persuasions and vintages. The walk-about is repeatedly puntuated by bursts of applause.
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writes: Lords Speaker Baroness Hayman use of poetry and prose analogy was brilliant. Obama was at his best.
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From the BBC's Ross Hawkins: The President of the United States gets a hug from Floella Benjamin. That's not something you see every day. The former children's TV presenter is now a Lib Dem peer and, it would appear, a fan.
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President Obama has now finally completed his exit from Westminster Hall. He is being driven back to Buckingham Palace. Later tonight, he and Mrs Obama will host a dinner for a number of distinguished guests, including the Queen.
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CNN reporter Zane Verjee says the speech was "orate, rhetorical, and eloquent". But according to the news organisation's live blog, some of the British press are not as impressed. "Not particularly memorable," one reporter has apparently said. "I think I preferred Dublin," said another.
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So we're "essential". Essential to Obama's re-election? Let's just say it's essential not to get carried away with this visit because we'll get essentally nothing from it.
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"This looks good back home," American journalist Jeff McAllister tells the BBC. He says many Americans still view Mr Obama as "an exotic" - 50% of Republicans still refuse to accept he was born on US soil - so seeing him against such a stately, historic backdrop can only help counter that.
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The BBC News channel's chief political correspondent Laura Kuenssberg says Downing Street sources are insisting that David Cameron and Barack Obama are absolutely on the same page when it comes to big issues. Nevertheless, she says, there are some differences between them on topics like economic recovery. Libya, a future Palestinian state.
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The Washington Post says the audience in Westminster Hall "showed much more reserve than the US Congress does during similar addresses, breaking into applause only once".
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I am sitting in the departure area at Midway airport, Chicago. President Obama's standing ovation in Westminster Hall is being shown live on all the TVs, and people are very impressed. A clear sign that Americans value the relationship.
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The BBC's diplomatic correspondent James Robbins says the UK has understood from the beginning that the US wanted to "front-load" its involvement in Libya before looking to Britain and France to take the lead.
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tweets An excellent address by President Barack Obama to MPs and Lords in Westminster Hall. Such an historic moment!
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Political analyst Professor Larry Sabato says the Obama visit has been kept off the airwaves and front pages in the US by the devastating storms in Missouri. But he also says the "specialness" of the UK-US relationship is so well-accepted and really, taken as read, that actually, it's not exceptional enough to warrant much coverage on the other side of the Atlantic.
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We're going to wrap up our live coverage here. Thank you very much for joining us and for all your contributions. The Obamas will host a dinner tonight for a roll-call of A-list guests, including the Queen, and you'll be able to hear more about that on the BBC News website, television and radio.
The News collected from : BBC News
