The pair signed what Trump 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, Trump said "we're starting that process", adding that it would begin "very, very quickly."

For his part, Kim said the two Cold War foes had vowed to "leave the past behind", pledging "the world will see a major change."

The extraordinary summit -- unthinkable only months ago -- comes after the two nuclear-armed foes appeared on the verge of conflict late last year as they slung personal insults and Kim conducted nuclear and missile tests.

It was the first-ever meeting between sitting leaders of the two countries and was closely watched around the world.

"We'll meet again," Trump said after a signing ceremony, standing with Kim on the verandah where they first met. "We will meet many times."

Trump said he "absolutely" would be willing to invite 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.

The two men shook hands for several seconds beneath the white-washed walls of an upscale hotel in neutral Singapore, Trump reaching out to touch the North Korean leader on his right shoulder.

As they sat down for their tete-a-tete, the US leader -- who had said he would know "within the first minute" if a deal would be possible -- predicted a "terrific relationship" with Kim.

In Seoul, South Korean President Moon Jae-in watched live on television, telling his ministers, he "could hardly sleep last night".

After huddling for around 40 minutes, Trump and Kim were joined by senior advisors before breaking for lunch, where prawn cocktail, short-rib confit, soy-braised cod, and vanilla ice-cream were among the options.

The imagery for the high-stakes meeting was undoubtedly positive and Kim Yong-hyun, professor at Dongguk University in Seoul said: "The atmosphere of the summit looks very good."

"It will be hard for this meeting to agree on specific deals but it carries considerable significance as a starting point," he said.

- 'Fire and fury' -

Some analysts have warned the summit could produce more style than substance and Kelly Magsamen, a former top Pentagon Asia specialist, said: "We'll see if Kim actually gives him anything more than superficial pleasantries and some good TV."

Critics said the mere fact of the meeting meant Trump was legitimising Kim, who critics say runs a police state where human rights are routinely trampled.

"It's a huge win for Kim Jong-un, who now -- if nothing else -- has the prestige and propaganda coup of meeting one-on-one with the president, while armed with a nuclear deterrent," said Michael Kovrig, Crisis Group Senior Adviser for North East Asia.

The warm words and positive optics seemed a different era from when Trump was threatening to rain down "fire and fury" on Pyongyang and Kim attacked Trump as a "mentally deranged US dotard", as he fired off a series of provocative weapons tests.

The Singapore summit is a potentially legacy-defining meeting for both men -- comparable to president Richard Nixon's 1972 visit to China, or Ronald Reagan's 1986 summit with Mikhail Gorbachev in Reykjavik.

But many agreements have been made in the past with North Korea that have later fallen apart.

On the table is the vexed question of denuclearisation -- a word that means vastly different things to the two parties.

It remains far from clear that Pyongyang is willing to give up its nukes -- weapons that the regime sees as its ultimate guarantee of survival.

And on the eve of the meeting, aides for both men were still scrambling to narrow yawning differences.

Secretary of State Mike Pompeo said Monday that the United States was willing to offer Pyongyang "unique" security guarantees if it denuclearises.

But the summit was more likely it is the start of a longer process of negotiation.

The discussions "will set the framework for the hard work that will follow", Pompeo said.