US President Donald Trump today described his first-ever historic meeting with North Korean leader Kim Jong-un as "really fantastic" and said they have agreed to "sign" an unspecified document after their "very positive" summit, aimed at normalising ties and complete denuclearisation of the Korean peninsula.

Trump and Kim met and shook hands at Capella Singapore hotel in Singapore's Sentosa Island after months of diplomatic twists and turns.

The two leaders had a one-on-one meeting, with translators only, followed by an expanded meeting including their top aides and a working lunch.

"A lot of progress. Really, very positive, I think better than anybody could have expected, top of the line, really good," Trump said after the lunch.

"We're going right now for a signing," Trump said. It was not clear what the two leaders would sign but Trump said the public will find out soon.

Earlier, the summit began with the two leaders meeting at the luxurious hotel.

They walked towards each other, stood face to face for the first time and shared a 12-second handshake against a backdrop of American and North Korean flags. They then appeared to share a few light-hearted words as they walked down a corridor to the hotel's library.

Trump and Kim are the first leaders of their respective countries to meet, marking the culmination of months of diplomatic wrangling and negotiations.

In brief remarks to the media before the start of their meeting, Trump hoped that the historic summit would be "tremendously successful."

Sitting next to the North Korean leader, the US President said, "We will have a terrific relationship ahead."

"I feel really great. It's gonna be a great discussion and I think tremendous success. I think it's gonna be really successful and I think we will have a terrific relationship, I have no doubt," Trump responded when asked how he felt in the first minute.

Kim then said there were a number of "obstacles" to the meeting taking place in Singapore today.

"We overcame all of them and we are here today," he told reporters through a translator.

Trump then said "Thank you very much".

The two leaders held one-on-one meeting for around 45-minutes with just translators present.

According to North Korean media, Kim actually arrived at the venue 7 minutes earlier than Trump to show respect as it's cultural, the young one should arrive earlier than the elder one. The red tie that Trump wears may also show some respect to Kim as well, it's the colour that North Koreans like.

Later, they emerged from their one-on-one meeting and headed into an expanded bilateral meeting.

When asked how the discussion went, Trump said: "Very, very good. Excellent relationship."

Kim was asked at least three times if he would give up his nuclear weapons. In response, he just smiled.

Trump said he believed he and Kim will "solve a big problem, a big dilemma" and that by working together, "we will get it taken care of".

"There will be challenges ahead but we will work with Trump. We overcame all kinds of skepticism and speculations about this summit and I believe that this is good for the peace," Kim responded.

After the delegation-level meeting, Trump joined Kim for a working lunch which had a mix of Western and Asian dishes, such as Korean stuffed cucumber and beef short rib confit with some Haagen-Dazs ice cream to top things off.

As the two men entered the room, photographs were taken, with Trump joking about wanting to get a "beautiful picture" in which he looked good.

The two leaders then took a post-lunch stroll in the courtyard of the hotel.

On the agenda of the meeting is North Korea's nuclear capabilities, which Trump is hoping to convince them to abandon in exchange for economic assistance.

On the eve of the summit, the US offered "unique" security guarantees to North Korea in return for a "complete, verifiable and irreversible" denuclearisation.

The summit at Sentosa - the first between a sitting US president and a North Korean leader - will mark a turnaround of relations between Trump, 71, and Kim, 34, after a long-running exchange of threats and insults.

Sentosa is a popular tourist island a few hundred metres off the main island of Singapore.

The US insists it will accept nothing less than complete denuclearisation of the Korean peninsula.

The official North Korean news agency said on Sunday that Kim was ready to talk about "denuclearisation" and a "durable peace" at a summit held "for the first time in history under the great attention and expectation of the whole world."

Trump said on Saturday that Kim has a "one-time shot" to make history.

"I feel that Kim Jong-un wants to do something great for his people," he had said.