Theresa May has welcomed Donald Trump to the UK with a lavish black-tie dinner - during which she made a case for a new trade deal with the US.
The US president and the first lady were given a red carpet reception at Blenheim Palace, Oxfordshire.
It is Mr Trump's first visit to the UK since he won the 2016 election.
Protesters gathered outside the US ambassador's residence in London, where the Trumps are staying tonight, and near Blenheim Palace.
Police estimated there were about 1,000 people demonstrating near the palace, the birthplace of wartime Prime Minister Winston Churchill.
His visit comes as the government published its plans for the UK's relations with the EU after Brexit.
Earlier this week Mr Trump said it was "up to the people" whether the prime minister stayed on after two cabinet ministers resigned over her Brexit policy.
Mrs May noted that more than one million Americans work for UK-owned firms, telling Mr Trump: "As we prepare to leave the European Union, we have an unprecedented opportunity to do more.
"It's an opportunity to reach a free trade agreement that creates jobs and growth here in the UK and right across the United States.
"It's also an opportunity to tear down the bureaucratic barriers that frustrate business leaders on both sides of the Atlantic.
"And it's an opportunity to shape the future of the world through co-operation in advanced technology, such as artificial intelligence."
'Defeat politics of division'
But Labour leader Jeremy Corbyn called Mr Trump "dangerous and inhumane", saying his policies were putting millions of lives at risk.
Criticising Mrs May for her warm welcome, he said: "Theresa May has invited President Trump to our country at a time when his dangerous and inhumane policies are putting the lives and wellbeing of millions of people at risk.
"We are committed to dialogue, including of course with those we strongly disagree with, and in government we would find a way to work with his administration while also standing up for our values.
"But instead the Tories are rewarding President Trump with a red carpet welcome.
"We must show Trump and the Tories that we can and we will defeat the politics of division."
The Trumps landed at Stansted Airport, Essex, at 13:50 BST before a helicopter took them to Winfield House in Regent's Park, where they are staying as guests of the US ambassador.
They stayed there for several hours before travelling to the 18th Century palace near Woodstock for the evening's events.
They were booed and jeered as their helicopter left Regent's Park, with protesters who had gathered to demonstrate against the president's visit shouting "shame on you" and banging pots and pans, in the hope he would hear them from behind the fence erected around the perimeter of Winfield House.
John Rees of the Stop the War group addressed the protesters, saying of Trump: "He's a wrecking ball for race relations, he's a wrecking ball for prosperity, he's a wrecking ball for women's rights, he's a wrecking ball for any peace and justice in this world and we have to stop him."
Mrs May and her husband Philip, joined by around 150 guests, waited outside Blenheim Palace in the sunshine for the president and first lady.
The Trumps were driven from their helicopter in the armoured presidential limousine, nicknamed The Beast, to the red carpet awaiting them.
They stood with the Mays to listen to the bandsmen of the Scots, Irish and Welsh Guards, who played the Liberty Fanfare, Amazing Grace and the National Emblem.
The president briefly held Mrs May's hand as they walked up the stairs to the palace, in a repeat of her visit to the White House when they held hands for a short time.
Business leaders from sectors including finance, travel, food and defence were among the guests at dinner.
Cabinet ministers including the newly-appointed Foreign Secretary Jeremy Hunt and Chancellor Philip Hammond were also present.
They were entertained by The Countess of Wessex's Orchestra, playing British and American hits of the 20th Century during dinner.
Only one person, the digital entrepreneur and philanthropist Baroness Martha Lane-Fox, has publicly refused the invitation.
They were given a best-of-British dinner, featuring Scottish salmon, Hereford beef fillet and strawberries with clotted cream ice-cream.
During the dinner, Mrs May said: "Mr President, Sir Winston Churchill once said that 'to have the United States at our side was, to me, the greatest joy'.
"The spirit of friendship and co-operation between our countries, our leaders and our people, that most special of relationships, has a long and proud history.
"Now, for the benefit of all our people, let us work together to build a more prosperous future."
'Dearest of friends'
She stressed that the US and UK are "not just the closest of allies, but the dearest of friends".
Perhaps in a reference to Mr Trump's meeting with Vladimir Putin on Monday, she said that Britain and the US work closely together in the interests of their shared security, "whether through targeting Daesh [Islamic State group] terrorists or standing up to Russian aggression".
On Friday, Mrs May and Mr Trump will go to watch a joint counter-terrorism exercise by British and US special forces at a military base.
The pair will then travel to Chequers - the PM's country residence in Buckinghamshire - for talks with the foreign secretary.
Extra security is in place to police the protests. More are planned for the second day of Mr Trump's visit.
The president and first lady will travel to Windsor on Friday afternoon to meet the Queen, before flying to Scotland to spend the weekend at Mr Trump's Turnberry golf resort. This part of the visit is being considered private.