The pair shook hands at a luxury hotel on Singapore's Sentosa island before proceeding to talks.
After the summit the leaders signed a "comprehensive" document, promising a new relationship between the nations.
They had been discussing defusing tensions and nuclear disarmament.
The document commits North Korea to work towards denuclearisation and promises "new... relations" between Washington and Pyongyang.
It makes no mention of US demands for "complete, verifiable and irreversible denuclearisation".
At a signing ceremony the leaders gave brief comments to the press but failed to elaborate their views on the document.
"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.