Previous White House administrations had denied Kim, his father and grandfather, the "reward" of a face-to-face meeting with a sitting United States president.
Kim, who met Singaporean Prime Minister Lee Hsien Loong on Sunday after arriving, thanked him for hosting the event.
Monday's meeting is apparently aimed at narrowing the gap between the US and North Korea over the demand for North Korea to give up its nuclear weapons.
As Trump was trying to make "fool trade" happen, Secretary of State Mike Pompeo tweeted a series of photos showing State Department officials meeting with foreign diplomats and generally behaving like normal White House officials headed into a high-stakes summit.
The clearest benchmarks about what the U.S.is seeking from the summit, however, have come from Pompeo rather than Trump.
Their meeting, which will be held at the city-state's Capella Hotel, will mark the first time a North Korean leader has met with a sitting U.S. president.
The two leaders are scheduled to meet for the first time on Tuesday at the Capella Hotel on Sentosa Island, where the process regarding denuclearisation of North Korea may kick-start in exchange for economic aid. Hopefully we will have liked each other and we'll start that process. Progress could lead to a peace treaty and formally end a state of war on the Korean peninsula, which has persisted for almost 70 years.
So the summit was almost derailed by talk from Trump's hawkish National Security Advisor John Bolton of a "Libya model" of disarmament.
Kim and Trump both arrived in Singapore a few hours apart on Sunday.
A large limousine with a North Korean flag could then be seen surrounded by other black vehicles with tinted window as it sped through the city's streets to the St Regis Hotel, where China's President Xi Jinping once stayed.
Singapore has spent $S20 million hosting the summit, half of that on security costs, with Lee saying it was "in our profound interest" to contribute to such an worldwide endeavour.
"This is a leader who is really an unknown personality", Trump said.
Is Kim Jong Un fluent in English?
During the meeting, the officials also celebrated Trump's birthday.
Mr Kim has only publicly left his country three times since taking power after his despot father's death in late 2011 - twice traveling to China and once across his shared border with the South to the southern part of the Demilitarised Zone in recent summits with the leaders of China and South Korea respectively.
Korean Studies Institute director David Kang said: "The most we can hope to get out of this is that Kim and Trump find that they can have some working relationship".
"Wide-ranging and profound views on the issue of establishing new DPRK-US relations, the issue of building a permanent and durable peace-keeping mechanism on the Korean Peninsula, the issue of actualizing the denuclearization of the Korean peninsula and other issues of mutual concern" will be discussed, state-run KCNA reported early Monday.
Earlier this morning, the tweeter-in-chief expressed his elation about being in Singapore and feeling "excitement in the air".