US president Donald Trump said on Tuesday that he has formed a "special bond" with North Korea's leader Kim Jong-un following a historic meeting during which the two men pledged to meet again.
Mr Trump and Mr Kim signed what the US president called a "pretty comprehensive" and "very important" document, which spoke of "new US-DPRK relations" and committed Washington to "security guarantees".
However, asked about denuclearisation — the crux of the summit, Mr Trump said "we're starting that process", adding that it would begin "very, very quickly".
According to photographs of the text, North Korea committed to "work towards" denuclearisation.
Mr Kim, in turn, said the two Cold War foes had vowed to "leave the past behind", pledging "the world will see a major change".
Mr Trump said after a signing ceremony that "we'll meeting again" as he stood with Mr Kim on the veranda where they first met. "We will meet many times."
Mr Trump said he "absolutely" would be willing to invite Mr Kim — whose regime has been criticised for widespread human rights abuses — to the White House.
Earlier, the pair shared warm words and a historic handshake as they sought to confront a nuclear stand-off and enmity stretching back decades.
Mr Trump had said at the start of the summit in Singapore that he forged a "good relationship" with the North Korean leader, while Mr Kim said the meeting was "a good prelude to peace".
Should they succeed in making a diplomatic breakthrough, it could bring lasting change to the security landscape of Far East Asia, like the visit of former US president Richard Nixon to China in 1972 led to the transformation of China.
For live updates on the Singapore summit, click here
Both men had looked serious as they got out of their limousines for the summit at the Capella hotel on Singapore's Sentosa, a resort island with luxury hotels, a casino, man-made beaches and a Universal Studios theme park.
But, with cameras of the world's press trained on them, they displayed an initial atmosphere of bonhomie as they met.
After a handshake, they were soon smiling and holding each other by the arm, before Mr Trump guided Mr Kim to the library where they held a meeting with only their interpreters. Mr Trump had said on Saturday he would know within a minute of meeting Mr Kim whether he would reach a deal.
Inside, they sat alongside each other against a backdrop of North Korean and US flags, with Mr Kim beaming broadly as the US president gave him a thumbs up.
The combatants of the 1950-53 Korean War are technically still at war, as the conflict, in which millions of people died, was concluded only with a truce.
After initial exchanges lasting around 40 minutes, Mr Trump and Mr Kim emerged, walking side-by-side through the colonnaded hotel before entering a meeting room, where they were joined by their most senior officials.
Mr Kim was heard telling Mr Trump through a translator: "I think the entire world is watching this moment. Many people in the world will think of this as a scene from a fantasy … science fiction movie."
Asked by a reporter how the meeting was going, Mr Trump said: "Very good. Very, very good. Good relationship."
Mr Kim also sounded positive about the prospects.
"We overcame all kinds of scepticism and speculations about this summit and I believe that this is good for the peace," he said. "I believe this is a good prelude for peace."
Mr Trump was joined by Secretary of State Mike Pompeo, National Security Adviser John Bolton, and John Kelly, White House Chief of Staff, for the expanded talks, while Mr Kim's team included former military intelligence chief Kim Yong Chol, foreign minister Ri Yong Ho and Ri Su Yong, vice chairman of the ruling Workers' Party.
As the two leaders met, Singapore navy vessels, and air force Apache helicopters patrolled, while fighter jets and an Gulfstream 550 early warning aircraft circled.
Financial markets were largely steady in Asia and did not show any noticeable reaction to the start of the summit. The dollar was at a three-week high and the MSCI index of Asia-Pacific shares was largely unchanged from Monday.
While Mr Trump and Mr Kim search each other’s eyes and words for signs of trust or deceit, the rest of the world will be watching, hoping that somehow these two unpredictable leaders can find a way to defuse one of the planet's most dangerous flashpoints.
A body language expert said both men tried to project command as they met, but also displayed signs of nerves.
After the meetings, the two teams met for a working lunch, where beef short ribs, sweet and sour pork and "Daegu Jormin", or Korean braised cod, were served for the main course, according to the menu. That was to be followed by dark chocolate tarts, pastries and vanilla ice cream for dessert.
In the hours before the summit began, Mr Trump expressed optimism about prospects for the first-ever meeting of sitting US and North Korean leaders, while Mr Pompeo injected a note of caution whether Mr Kim would prove to be sincere about his willingness to denuclearise.
Officials of the two sides held last-minute talks to lay the groundwork for the summit of the old foes, an event almost unthinkable just months ago, when they were exchanging insults and threats that raised fears of war.
Staff-level meetings between the United States and North Korea were going "well and quickly", Mr Trump said in a message on Twitter on Tuesday.
But he added: "In the end, that doesn't matter. We will all know soon whether or not a real deal, unlike those of the past, can happen!"
Mr Pompeo said earlier the summit should set the framework for "the hard work that will follow", insisting that North Korea had to move toward complete, verifiable and irreversible denuclearisation.
North Korea, however, has shown little appetite for surrendering nuclear weapons it considers vital to the survival of Mr Kim's dynastic rule.
Sanctions on North Korea would remain in place until that happened, Mr Pompeo said on Monday. "If diplomacy does not move in the right direction … those measures will increase."
The White House said later that discussions with North Korea had moved "more quickly than expected" and Mr Trump would leave Singapore on Tuesday night after the summit, rather than Wednesday, as scheduled earlier.
Mr Kim is due to leave on Tuesday afternoon, a source involved in the planning of his visit has said.
One of the world's most reclusive leaders, Mr Kim visited Singapore's waterfront on Monday, smiling and waving to onlookers, adding to a more affable image that has emerged since his April summit with South Korean leader Moon Jae-in.
Mr Trump spoke to Mr Moon and Japan's Prime Minister Shinzo Abe, both key allies of Washington in the region, to discuss developments before the summit.
"I too, got little sleep last night," Mr Moon told his cabinet in Seoul as the summit began in Singapore.
"I truly hope it will be a successful summit that will open a new age for the two Koreas and the United States and bring us complete denuclearisation and peace."