Trump traveled to Singapore from Canada, where he a meeting with other world leaders.
Hours earlier, a jet carrying Kim landed.
Following the lunch, Singapore's Minister for Foreign Affairs Vivian Balakrishnan tweeted out a picture showing the president was gifted with a birthday cake.
North Korea's leader Kim Jong Un shakes hands with US President Donald Trump at the start of their historic US-North Korea summit, at the Capella Hotel on Sentosa island in Singapore. The North has faced crippling diplomatic and economic sanctions as it has advanced development of its nuclear and ballistic missile programs.
"It is a cost that we're willing to bear to play a small part in this historic meeting", the defense minister said.
Trump is set to meet with Lee on Monday. The official was not authorized to speak publicly about internal deliberations and insisted on anonymity. The North Korean leader's earlier arrival was reportedly a sign of respect to the older USA president. Trump previously said he would be able to tell in one minute if Kim was honest about giving up his nuclear weapons. But it will only be the beginning of a longer process.
Trump left his hotel first but arrived about seven minutes after Kim. American diplomats also work closely with Singapore on many strategic issues, and it has been a longtime USA trading partner and ally.
U.S. and North Korean officials are set to meet Monday morning in Singapore to make final preparations for Tuesday's meeting.
Trump was joined by Pompeo, National Security Adviser John Bolton, and John Kelly, White House Chief of Staff, for the expanded talks, while Kim's team included former military intelligence chief Kim Yong Chol, foreign minister Ri Yong Ho and Ri Su Yong, vice chairman of the ruling Workers' Party. It's a reflection of the intense global curiosity over Kim's sudden turn to diplomacy in recent months after a slew of North Korean nuclear and missile tests past year raised serious fears of war.
"I truly hope it will be a successful summit that will open a new age for the two Koreas and the United States and bring us complete denuclearisation and peace".
In the weeks leading up to the meeting, several locations were considered: the Korean Demilitarized Zone, or DMZ, the border barrier that separates North and South Korea; Ulaanbaatar, Mongolia; Geneva, Switzerland; and Stockholm, Sweden.
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After a handshake, they were soon smiling and holding each other by the arm, before Trump guided Kim to the library where they held a meeting with only their interpreters.
President Donald Trump waves from inside his auto as he heads into the Istana Presidential Palace, where he will meet with Singapore Prime Minister Lee Hsien Loong, 11 June 2018.
Trump holds up a document signed by him and Kim.
Nor was there any reference to finally signing a peace treaty.
It's unclear what Trump and Kim might decide Tuesday. Pyongyang's weapons program benefits from forced labor.
He called Kim "very smart" and a "very worthy, very hard negotiator".
Any nuclear deal will hinge on North Korea's willingness to allow unfettered outside inspections of the country's warheads and nuclear fuel, much of which is likely kept in a vast complex of underground facilities.
The North Korea leader said through a translator that the document "heralds a new start, leaving the past behind".
The fighting ended on 27 July, 1953, but the war technically continues today because instead of a difficult-to-negotiate peace treaty, military officers for the USA -led United Nations, North Korea and China signed an armistice that halted the fighting.
Mr Trump said he was feeling "really great" and Mr Kim said North Korea and the U.S. had overcome obstacles to be able to hold talks, saying it "was not easy to get here".