Mr Trump descended from Air Force One into the steamy Singapore night, greeting officials and declaring he felt "very good" before being whisked away to his hotel via a route lined with police and photo-snapping onlookers.
Mr Trump travelled to Singapore from Canada, where he met with other world leaders.
Hours earlier, a jet carrying Mr Kim landed. After shaking hands with Singapore's foreign minister, Mr Kim sped through the streets in a limousine, two large North Korean flags fluttering on the car, surrounded by other black vehicles with tinted windows and bound for the luxurious and closely guarded St Regis Hotel.
He and Mr Trump are set to meet on Tuesday morning in the first summit of its kind between a leader of North Korea and a sitting US president.
A US official said on Monday that Mr Trump and Mr Kim will first meet one on one with translators in a session that could last up to two hours before they open the meeting to their respective advisers.
Mr Kim smiled broadly on Sunday evening as he met with Singapore prime minister Lee Hsien Loong.
"The entire world is watching the historic summit between (North Korea) and the United States of America, and thanks to your sincere efforts... we were able to complete the preparations for the historic summit," Mr Kim told Mr Lee through an interpreter.
Mr Trump is set to meet with Mr Lee on Monday.
Mr Trump has said he hopes to make a legacy-defining deal for the North to give up its nuclear weapons, though he has recently sought to minimise expectations, saying more than one meeting may be necessary.
The North has faced crippling diplomatic and economic sanctions as it has advanced development of its nuclear and ballistic missile programmes.
Experts believe the North is on the brink of being able to target the entire US mainland with its nuclear-armed missiles, and while there is deep scepticism that Mr Kim will quickly give up those hard-won nukes, there is also some hope that diplomacy can replace the animosity between the US and the North.
US and North Korean officials are set to meet on Monday morning in Singapore to make final preparations for Tuesday's meeting.
White House press secretary Sarah Huckabee Sanders said Sung Kim, the US 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.
The North Korean autocrat's every move will be followed by 3,000 journalists who have converged on Singapore, and by gawkers around the world, up until he shakes hands with Mr Trump on Tuesday.
It is a reflection of the intense global curiosity over Mr Kim's sudden turn to diplomacy in recent months after a slew of North Korean nuclear and missile tests last year raised serious fears of war.
But it was only Monday morning in North Korea that the government news agency reported that Mr Kim was in Singapore, had met with Prime Minister Lee Hsien Loong and would meet Mr Trump on Tuesday.
One dispatch by the Korean Central News Agency said North Korea and the US would exchange "wide-ranging and profound views" on establishing new relations, building a "permanent and durable peace-keeping mechanism," achieving denuclearisation and "other issues of mutual concern, as required by the changed era".
Part of the interest in Tuesday's summit is simply because Mr Kim has had limited appearances on the world stage. He has only publicly left his country three times since taking power after his father's death in late 2011 - travelling twice to China and once across his shared border with the South to the southern part of the Demilitarised Zone for recent summits with the leaders of China and South Korea, respectively.
But it is Mr Kim's pursuit of nuclear weapons that gives his meeting with Mr Trump such high stakes.
The meeting was initially meant to rid North Korea of its nuclear weapons, but the talks have been portrayed by Mr Trump in recent days more as a get-to-know-you session.
Mr Trump has also raised the possibility of further summits and an agreement ending the Korean War by replacing the armistice signed in 1953 with a peace treaty. China and South Korea would have to sign off on any legal treaty.
It is unclear what Mr Trump and Mr Kim might decide on Tuesday.
Pyongyang has said it is willing to deal away its entire nuclear arsenal if the United States provides it with reliable security assurances and other benefits. But many say this is highly unlikely, given how hard it has been for Mr Kim to build his programme and given that the weapons are seen as the major guarantee to his holding onto unchecked power.
Any nuclear deal will hinge on North Korea's willingness to allow unfettered outside inspections of the country's warheads and nuclear fuel, much of which is likely kept in a vast complex of underground facilities. Past nuclear deals have crumbled over North Korea's reluctance to open its doors to outsiders.
Another possibility from the summit is a deal to end the Korean War, which North Korea has long demanded, presumably, in part, to get US troops off the Korean Peninsula and, eventually, pave the way for a North Korean-led unified Korea.
The fighting ended on July 27 1953 but the war technically continues today because instead of a difficult-to-negotiate peace treaty, military officers for the US-led United Nations, North Korea and China signed an armistice that halted the fighting.
The North may see a treaty - and its presumed safety assurances from Washington - as its best way of preserving the Kim family dynasty. The ensuing recognition as a "normal country" could then allow sanctions relief, and later international aid and investment.
Just meeting with Mr Trump will also give Mr Kim a recognition North Korea has long sought, setting him up as global player and the leader of a country worthy of respect.