The event at Sentosa Island in Singapore, starting around 9 a.m. Tuesday local time, was a sharp reversal from the rhetoric between Trump and Kim for much of previous year, as each mercurial figure engaged in name calling and provocative rhetoric.
It is also unclear whether USA officials will raise human rights issues in meetings with the North Koreans.
The US president has also said that if he thinks things are going badly, he will walk out of the meeting, but if things go well, Mr Kim could receive an invitation to the White House.
U.S. Secretary of State Mike Pompeo speaks during a press briefing, June 11, 2018 in Singapore one day before President Donald Trump will meet with North Korean leader Kim Jong Un.
The stakes are high - the USA seeks nothing less than the complete and verifiable denuclearization of the Korean peninsula, and Kim wants to see the sanctions on the North eased. Recently, Kim Jong Un sent him to the United States to hand-deliver a letter to Trump, making him North Korea's most senior official to visit its wartime foe in 18 years. Trump traveled to Singapore from Canada, where he attended a meeting with other world leaders.
A few minutes later, as he sat with Kim for the meeting, Trump told reporters, "We will have a terrific relationship, I have no doubt". He told reporters he thinks he will know nearly immediately whether a deal can be made, saying: "I will know, just my touch, my feel".
North Korean Leader Kim Jong Un (centre) touring Singapore's Marina Bay Area overlooking the city-state's skyline.
There was no word on how Trump was spending the night before the summit.
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Kim Yong Chol's rise has baffled many North Korea watchers because he is the ruling Workers' Party official in charge of South Korea ties, not global or US relations.
He was received by Singapore's Foreign Affairs Minister Vivian Balakrishnan.
Trump is hoping to get Kim to agree to get rid of his nuclear weapons, though he has said it may take many more meetings to get to that point.
Trump advisers cast his actions as a show of strength before the Kim meeting.
For both men, the summit, the first between leaders of North Korea and the United States, is likely to be a defining moment of their careers.
He said about half of the cost is for security.
It's so far unclear whether the US and North Korea will release some kind of joint statement or agreement following the sit-down.
He went out of his way to be cordial during two summits with South Korean President Moon Jae-in in April and May.
North Korea has previously said it was willing to discuss denuclearization, but experts have expressed concern over the varying definitions of the term held by Washington and Pyongyang.