But where are they meeting, and what have they been doing?
The White House has released a schedule for Tuesday’s events. Here’s what they have done so far, and what’s coming up next…
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8am Singapore time (1am UK time)
President Donald Trump departed the Shangri-La Hotel and started making his way to the Capella Hotel, where the summit is taking place. He arrived about 20 minutes later – followed moments later by Mr Kim.
Out of the dark – North Koreans told of summit
9am Singapore time (2am UK time)
Mr Trump and Mr Kim met each other for the very first time at the hotel on Sentosa Island in Singapore.
The pair shook hands and exchanged pleasantries – later sitting in front of a backcloth of US and North Korean flags.
While the US president predicted the pair will have a terrific relationship, Mr Kim said The road leading up to here was not that smooth. We’ve overcome so many obstacles which blocked our eyes and ears, dragging our feet down due to the past history, prejudices. But despite of all that, we’ve made it here.
Trump confident of terrific relationship with Kim
9.15am Singapore time (2.15am UK time)
Both leaders participated in a one-to-one bilateral meeting, lasting for about 45 minutes. This was followed by an extended bilateral meeting involving other officials.
They included US secretary of state Mike Pompeo, chief of staff John Kelly and national security adviser John Bolton.
Mr Trump described their private tete-a-tete as very, very good – and, as they discussed nuclear matters, the president predicted that together they will solve a big problem, a big dilemma.
11.30am Singapore time (4.30am UK time)
Mr Trump and Mr Kim have a working lunch – and the menu has been revealed.
Starters included a traditional prawn cocktail, green mango kerabu with a honey lime dressing and fresh octopus, as well as Korean stuffed cucumber.
For the main course, the undoubtedly peckish leaders tried beef short rib confit with potato dauphinoise and steamed broccolini, a combination of sweet and sour crispy pork and Yangzhou fried rice, and daegu jorim – soy braised codfish with radish and Asian vegetables.
On offer for dessert was dark chocolate tartlet ganache, tarte tropezienne and Häagen-Dazs vanilla ice cream with cherry coulis.
White House press secretary Sarah Huckabee Sanders joined them for lunch, as did ambassador Sung Kim and US national security council senior director for Asia Matt Pottinger.
4pm Singapore time (9am UK time)
The US president will participate in a media availability following the conclusion of talks with his North Korean counterpart – possibly even a news conference.
6.30pm Singapore time (11.30am UK time)
Mr Trump will leave the Capella Hotel and travel to Paya Lebar Air Base. He will fly onwards to Andersen Air Force Base in Guam, Joint Base Pearl Harbour and Joint Base Andrews – arriving at 3am UK time on Wednesday morning.
News Source SkyNews