Kim has offered to deal away his nuclear arsenal but doubts remain over whether he is truly willing to give up his nukes or is using the summit to weaken US-led sanctions against his country.
The two men are expected to meet on their own for the better part of an hour, with only a pair of interpreters in the room.
Concerns have been raised over how the monumental talks, which began shortly after they met each other for the first time, took place without anyone on hand to take notes or witness what was being said.
But after a flurry of diplomatic overtures eased tension in recent months, the two leaders are now headed for a history-making handshake that US officials hope could eventually lead to the dismantling of a North Korean nuclear programme that threatens the United States. Kim then called Trump a "mentally deranged U.S. dotard" and a "frightened dog".
While Trump was optimistic about prospects for the first-ever meeting of sitting U.S. and North Korean leaders, U.S. Secretary of State Mike Pompeo injected a note of caution, saying it remained to be seen if Kim was honest about his willingness to denuclearise.
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USA and global markets rose modestly on Monday, as investors made preparations for President Donald Trump's meeting with North Korean leader Kim Jong Un in Singapore. Analysts say Kim's diplomatic outreach in recent months after a flurry of nuclear and missile tests in 2017 indicates he is eager for sanctions relief to build his economy and the worldwide legitimacy the summit with Trump would provide. She previously worked as an English-language interpreter for a North Korean delegation attending now-stalled six-nation nuclear disarmament talks.
The PBS NewsHour's Nick Schifrin tells managing editor and anchor Judy Woodruff the latest from Singapore.
"Why would Kim... believe any commitments President Trump makes when he arbitrarily tears up an agreement with which the other party is complying?" said Antony Blinken, who was deputy secretary of state under Barack Obama.
The KCNA news agency said the two leaders will exchange "wide-ranging and profound views" during their meeting at the five-star Capella Hotel on Sentosa Island.
Later in the day, advisers joined the talks for a larger bilateral session and a working lunch. Beyond the impact on both leaders' political fortunes, the summit could shape the fate of countless people - the citizens of impoverished North Korea, the tens of millions living in the shadow of the North's nuclear threat, and millions more worldwide.
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After a flurry of diplomatic overtures that eased tension, the leaders are headed for a history-making handshake that US officials hope could eventually lead to the dismantling of a North Korean nuclear programme that threatens the United States.
Iizuka spoke highly of Trump's promise to include the abduction issue in the summit agenda along with the denuclearization of North Korea, hoping the return of the abductees will be ensured at the U.S.
Kim and Trump arrived in Singapore on Sunday.
The young North Korean leader is staying under heavy guard at a city center hotel and had not appeared in public on Monday until about 9 p.m., when one of his security officials buzzing about the hotel lobby shouted "It's all ready!" Mr Kim is reported to be flying out even earlier, at 14:00 local time (6pm NZT).
Kim, one of the world's most reclusive leaders, made an evening tour of sites on Singapore's waterfront, on the eve of the summit due to start on Tuesday at a nearby resort island.
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The North has faced crippling diplomatic and economic sanctions as it has advanced development of its nuclear and ballistic missile programs.