If the joint statement does lead to a lasting detente, it could fundamentally change the security landscape of Northeast Asia, just as former US President Richard Nixon's visit to Beijing in 1972 led to the transformation of China.
"We had a really fantastic meeting, a lot of progress, very positive", Trump said.
Bringing U.S. troops home from North Korea was "not a part of the equation right now", Mr. Trump said when asked for specifics about what he had agreed to. Trump then directed Kim to walk down a hallway, where they briefly spoke.
They met one on one for about 40 minutes, had an expanded meeting joined by a few aides, and concluded their morning activities with a working lunch. "And we'll see how far we get, but I'm very optimistic", he continued. I thought I was going to play basketball and meet the people and that was it.
"Very good, very good".
Mr Lee also wished Mr Kim success for the summit, and expressed the hope that it would improve prospects for peace and stability on the Korean peninsula and in the wider region.
He flew into Singapore late Monday to be present for the historic first summit between the two leaders.
Reuters quoted body language experts saying that despite the cordial words, the pair exhibited subconscious signs of nerves.
The two clasped hands for a long while Tuesday as they posed for photos in front of a row of US and North Korean flags.
For each leader, the summit represented a major gamble, and the outcome will be dissected around the globe for a sense of whether one of the world's greatest security threats - Kim's nuclear arsenal - can be eliminated.
But he added: "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!" he wrote. "But we overcame all of them and we are here today".
Setting out the United States position before the summit, Pompeo stressed that the Trump administration would only accept complete denuclearisation of the North.
The leaders will meet on Tuesday on the resort island of Sentosa and talks are expected to centre on ending the North's nuclear weapons and missile programmes in return for diplomatic and economic incentives.
Kim has since left the island.
When Singaporean Prime Minister Lee Hsien Loong visited the worldwide media center for the summit at F1 Pit Building on Sunday, he said it will cost the city state "around 20 million Singaporean dollars" (W16.1 billion, US$1=W1,078) to host the "historic summit".