SINGAPORE: President Donald Trump and North Korea's Kim Jong Un concluded an extraordinary summit Tuesday with lofty promises by the American president to take care of a “very dangerous problem” and Kim forecasting “major change for the world”.

They signed a document that Trump described as “pretty comprehensive”, but he declined to describe it, saying the details would be revealed later.

The document signing followed a series of meetings at a luxury Singapore resort.

Meeting with staged ceremony on a Singapore island, Trump and Kim came together for a summit that seemed just unthinkable months ago, clasping hands in front of a row of alternating US and North Korean flags, holding a one-on-one meeting, additional talks with advisers and a working lunch.

Throughout the summit that could chart the course for historic peace or raise the spectre of a growing nuclear threat, both leaders expressed optimism. Kim called the sit-down a “good prelude for peace” and Trump pledged that “working together we will get it taken care of.”

In advance of their private session, Trump predicted “tremendous success” while Kim said through an interpreter that “we have come here after overcoming” obstacles.

Aware that the eyes of the world were on a moment many people never expected to see, Kim said many of those watching would think it was a scene from a “science fiction movie”.

In the run-up to the meeting, Trump had predicted the two men might strike a nuclear deal or forge a formal end to the Korean War in the course of a single meeting or over several days. But in the hours before the summit, the White House unexpectedly announced Trump would depart Singapore earlier than expected -- Tuesday evening -- raising questions about whether his aspirations for an ambitious outcome had been scaled back.

Giving voice to the anticipation felt around the world, South Korean President Moon Jae-in said Tuesday he “hardly slept” before the summit. Moon and other officials watched the live broadcast of the summit before a South Korean Cabinet meeting in his presidential office

The meeting was the first between a sitting US president and a North Korean leader.

After meeting privately and with aides, Trump and Kim moved into the luncheon at a long flower-bedecked table. As they entered, Trump injected some levity to the day's extraordinary events, saying: “Getting a good picture everybody? So we look nice and handsome and thin? Perfect.”

Then they dined on beef short rib confit along with sweet and sour crispy pork.

And as they emerged from the meal for a brief stroll together, Trump appeared to delight in showing his North Korean counterpart the interior of “The Beast”, the famed US presidential limousine known for its high-tech fortifications.

Critics of the summit leapt at the leaders' handshake and the moonlight stroll Kim took Monday night along the glittering Singapore waterfront, saying it was further evidence that Trump was helping legitimise Kim on the world stage as an equal of the US president. Kim has been accused of horrific rights abuses against his people.

Trump responded to such commentary on Twitter, saying: “The fact that I am having a meeting is a major loss for the US, say the haters & losers.” But he added “our hostages” are back home and testing, research and launches have stopped.

Trump also tweeted: “Meetings between staffs and representatives are going well and quickly ... but 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!”

The summit capped a dizzying few days of foreign policy activity for Trump, who shocked US allies over the weekend by using a meeting in Canada of the Group of Seven industrialised economies to alienate America's closest friends in the West. Lashing out over trade practices, Trump lobbed insults at his G-7 host, Canadian Prime Minister Justin Trudeau. Trump left that summit early and, as he flew to Singapore, tweeted that he was yanking the US out of the group's traditional closing statement.

The optimistic summit was a remarkable change in dynamics from less than a year ago, when Trump was threatening “fire and fury” against Kim, who in turn scorned the American president as a “mentally deranged US dotard”.Beyond the impact on both leaders' political fortunes, the summit could shape the fate of countless people _ the citizens of impoverished North Korea, the tens of millions living in the shadow of the North's nuclear threat, and millions more worldwide.

Experts believe the North is close to being able to target the entire US mainland with its nuclear-armed missiles, and while there's deep skepticism that Kim will quickly give up those hard-won nukes, there's also some hope that diplomacy can replace the animosity between the US and the North.