Shortly after arriving in Singapore in the afternoon yesterday, Mr Kim, who is Chairman of the State Affairs Commission in North Korea, the nation's highest decision-making body, met Prime Minister Lee Hsien Loong for the first time.
Rodman also goes back years with Trump and appeared on his "Celebrity Apprentice" show.
Mr Trump said: "We had a terrific day and we learned a lot about each other and our countries". Korean Central News Agency (KCNA) mentioned that the summit is being held "under the great attention and expectation of the whole world".
In the hours before the summit began, Trump expressed optimism about prospects for the first-ever meeting of sitting U.S. and North Korean leaders, while Pompeo injected a note of caution whether Kim would prove to be honest about his willingness to denuclearise. The language of the agreement involves North Korea committing to complete denuclearization and long-lasting peace in exchange for unspecified "security guarantees" from the United States.
About 5,000 journalists are in Singapore for the occasion, but only a handful of American and North Korean reporters and photographers were permitted at the venue when the two leaders greet each other.
The pair made brief remarks as they sat together before the world's media as the historic summit got under way, with the USA leader insisting "we will have a terrific relationship I have no doubt".
"I also think I'll know whether or not it will happen fast.it may not".
This is the furthest Kim is known to have traveled since he came to power in 2011 and there has never previously been a meeting between a US president and a North Korean leader.
Inside, they sat alongside each other against a backdrop of North Korean and USA flags, with Kim beaming broadly as the United States president gave him a thumbs up.
Just meeting with Trump will also give Kim recognition as the leader of a "normal" country and as an equal of the USA leader.
The arrival of the Air China Airbus 330 flight at Changi airport raised speculation that a delegation of North Korean officials had joined an advance team headed by Kim's close aide, Kim Chang-son, who was already in Singapore to prepare for the talks.
For each leader, the summit represented a major gamble, and the outcome will be dissected around the globe for a sense of whether one of the world's greatest security threats - Kim's nuclear arsenal - can be eliminated.
Trump wants the complete, verifiable and irreversible dismantling of North Korea's nuclear weapons program.
He said if diplomacy fails to move in the right direction, the sanctions "will increase".
Setting out the U.S. position before the summit, Pompeo stressed that the Trump administration would only accept complete denuclearisation of the North.