The pair shook hands at a luxury hotel in Singapore’s Sentosa island before proceeding to talks.
At the end of the summit the leaders signed a “comprehensive” document but did not explain what they had agreed.
They had been discussing defusing tensions and nuclear disarmament.
At a signing ceremony the leaders gave brief comments to the press but failed to specify what the document contained.
“I think both sides will be very impressed with the result,” Mr Trump said.
The US president said more details would be provided later. A press conference is scheduled for 16:00 local time (08:00 GMT).
Mr Trump spoke of a “special bond” with the North Korean leader and said he was “absolutely” willing to invite him to the White House.
“We’ve decided to leave the past behind,” Mr Kim said. “The world will see major changes.”
Still, analysts remain divided on what the summit will ultimately achieve. Some see it as a propaganda win for Mr Kim, others a potential path to peace.
How the historic moment unfolded
The summit began with a striking image, unimaginable just months ago.
The two men walked towards each other and firmly gripped each other’s hands in front of US and North Korean flags.
Mr Trump patted Mr Kim’s arm as they stood on red carpet and exchanged a few words before turning to face the gathered press.
“I feel really great. We’re going to have a great discussion and will be tremendously successful,” the US president said.
Sitting alongside each other, ahead of a one-on-one meeting, the pair appeared relaxed against the odds.
“It was not easy to get here,” Mr Kim said. “There were obstacles but we overcame them to be here,” and his counterpart responded “That’s true.”
The two men, accompanied only by interpreters, spoke for a little under 40 minutes. They were then joined by small delegations of advisors for a working lunch.
From Washington the group included US Secretary of State Mike Pompeo and security adviser John Bolton, and on Pyongyang’s side, Mr Kim’s “right-hand man” and former spy chief Kim Yong-chol, and foreign minister Ri Yong-ho.
Over lunch they shared a mix of Western and Korean dishes, including stuffed cucumbers and Daegu jorim, a soy-braised fish dish.