This is the absolute crux of the summit and there remains a chasm between the two sides that diplomats were scrambling to close on the eve of the meeting.
For his part, Kim hailed the "historic meeting" and said they "decided to leave the past behind".
"Wide-ranging and profound views on the issue of establishing new DPRK-U.S. relations, the issue of building a permanent and durable peace-keeping mechanism on the Korean Peninsula, the issue of realizing the denuclearization of the Korean peninsula and other issues of mutual concern, as required by the changed era, will be exchanged at the DPRK-U.S. summit talks", the KCNA reported in English.
"If the summit becomes a success, the Singaporean efforts will go down in history", Kim said.
Kim brought a significant security presence to Singapore, including three airplanes and a convoy of armored vehicles. They believe Kim's latest engagement is aimed at getting the United States to ease the crippling sanctions that have squeezed the impoverished country.
Meanwhile, the US President should arrive in Singapore in the coming hours from Canada, where he left the G7 Summit due to disagreements with the rest of the members of that bloc.
Siegfried Hecker, a noted USA nuclear expert, said the immediate CVID of the North was "unimaginable" and "tantamount to a North Korean surrender scenario". Lee complimented the bold and admirable decision by Kim and Trump to come together for the summit.
"If I think it won't happen, I'm not going to waste my time", he said.
"Great to be in Singapore, excitement in the air!"
Trump's two-day stop in Quebec for a meeting with the leaders of the Group of Seven industrial nations was exactly the fiasco many feared.
The summit destination is Capella Hotel on Singapore's Sentosa Island.
Trump said Saturday that he believes he will know within the first minute whether Kim is seriously considering eliminating his nuclear arsenal and infrastructure, as the US demands.
The Trump administration has previously insisted on "complete, verifiable, irreversible disarmament" (CVID) of North Korea, with the emphasis on unilateral steps by Pyongyang rewarded by USA security assurances.
Washington hopes the summit will kick-start a process that eventually sees Mr Kim give up his nuclear weapons.
In March, Mr Trump surprised the world by accepting an invitation from Mr Kim - delivered via Seoul - to meet in person. He described it as a "get to know you plus" meeting.
Mr Kim arrived on a plane loaned from China along with his outspoken Foreign Minister, Ri Yong-ho, Defence Minister No Kwang-chol and his sister Kim Yo-jong.
Led by Sung Kim, a longtime State Department diplomat who now serves as the USA ambassador to the Philippines, the US team had held at least five sessions with the Pyongyang delegation over the past two weeks at the demilitarized zone in Korea.
White Housepress secretary Sarah Huckabee Sanders says the U.S. ambassador to the Philippines, who has taken the lead on policy negotiations with the North, will hold a working group with a North Korean delegation at 10 a.m. Monday local time, or 10 p.m. Sunday on the U.S. East Coast.
So the summit was almost derailed by talk from Trump's hawkish National Security Advisor John Bolton of a "Libya model" of disarmament. Libya unilaterally surrendered its nuclear weapons programme in 2003, but its leader, Muammar Gaddafi, was killed in 2011 by NATO-backed rebels.