Kim and Trump shook hands and smiled briefly in front of a row of their national flags at the Metropole hotel in the Vietnamese capital, Hanoi, before heading to dinner together.
Trump told reporters he thought the talks would be very successful, and asked if he was "walking back" on denuclearisation, said "no".
At their historic first summit in Singapore last June, Trump and Kim pledged to work toward denuclearisation and permanent peace on the Korean peninsula but little progress has been made.
Kim said they had overcome obstacles to hold their second summit and praised Trump for his "courageous decision" to begin a dialogue.
"Now that we're meeting here again like this, I'm confident that there will be an excellent outcome that everyone welcomes, and I'll do my best to make it happen," Kim said.
Trump and Kim held a 20-minute, one-on-one chat before sitting down to dinner with U.S. Secretary of State Mike Pompeo Trump's acting Chief of Staff Mick Mulvaney, Kim's top envoy, Kim Yong Chol, and North Korean Foreign Minister Ri Yong Ho.
On Thursday, the two leaders will hold a series of meetings, the White House said. The venue has not been announced.
"We’re going to have a very busy day tomorrow ... Probably a very quick dinner," said a smiling, relaxed looking Trump, seated beside Kim at a round table with the other four officials and two interpreters.
"Our relationship is a very special relationship."
Kim also appeared at ease. “We’ll have a very interesting dialogue," he told Trump.
Trump said late last year he and Kim "fell in love" but whether the bonhomie can move them beyond summit pageantry to substantive progress on eliminating Pyongyang's nuclear arsenal that threatens the United States is the question that will dominate the talks.
Trump and Kim's Singapore summit, the first meeting between a sitting U.S. president and a North Korean leader, ended with great fanfare but little substance over how to dismantle North Korea's nuclear weapons and ballistic missiles.
In the run-up to this summit, Trump has indicated a more flexible stance, saying he was in no rush to secure North Korea's denuclearisation. He repeated that on Wednesday, saying while some people believed the talks should be moving more quickly, he was satisfied.
Trump has held out the prospect of easing sanctions if North Korea does something "meaningful".
Some critics have said Trump appeared to be wavering on a long-standing U.S. demand for complete and irreversible denuclearisation by North Korea, and risked squandering leverage if he gave away too much, too quickly.
Asked if he would declare a formal end to the Korean War, which North Korea has long called for, Trump said: “We’ll see."
North and South Korea have been technically at war since their 1950-53 conflict, with the Americans backing the South, ended in a truce, not a treaty.
U.S. intelligence officials have said there is no sign North Korea will ever give up its entire arsenal of nuclear weapons, which it sees as its guarantee of national security. Analysts say it won't commit to significant disarmament unless punishing U.S.-led economic sanctions are eased.
The two sides have discussed specific and verifiable denuclearisation measures, such as allowing inspectors to observe the dismantlement of North Korea's Yongbyon nuclear reactor, US and South Korean officials say.
US concessions could include opening liaison offices or clearing the way for inter-Korean projects.
Despite little progress on his goal of ridding North Korea of its weapons programmes, Trump appeared to be betting on his personal relationship with North Korea's young leader, and the economic incentive after 70 years of hostility between their countries.
"Vietnam is thriving like few places on earth. North Korea would be the same, and very quickly, if it would denuclearize," Trump said on Twitter ahead of the meeting.
"The potential is AWESOME, a great opportunity, like almost none other in history, for my friend Kim Jong Un. We will know fairly soon - Very Interesting!"
Any deal will face scrutiny from American lawmakers and other sceptics who doubt North Korea is willing to give up the weapons.
For Trump, a deal that eases the North Korean threat could hand him a big foreign-policy achievement in the midst of domestic troubles.
While he is in Hanoi, his former personal lawyer Michael Cohen is testifying before U.S. congressional committees, with the president's business practices the main focus.
Cohen, in wide-ranging testimony he is due to deliver on Wednesday, refers to a comment Trump made to him about avoiding the US military draft for the Vietnam War on medical grounds: "'You think I'm stupid, I wasn't going to Vietnam'," Cohen cited Trump as saying.
"I find it ironic, President Trump, that you are in Vietnam right now," Cohen said in a draft statement seen by Reuters.
Trump, responding to the statement on Twitter, said Cohen was lying to reduce his prison time. He declined to respond when a reporter asked him about Cohen later.
This story has been published from a wire agency feed without modifications to the text. Only the headline has been changed.