North Korean leader Kim Jong-un and Donald Trump met in Vietnam for a second summit that the US hopes will persuade North Korea to give up its nuclear weapons in exchange for promises of peace and development.
The US president tweeted after a dinner with Mr Kim at Hanoi's French-colonial-era Metropole hotel that they had had "Great meetings" and a "Very good dialogue," while the White House said the two planned to sign a "joint agreement" after further talks on Thursday.
Mr Kim and Mr Trump shook hands and smiled briefly in front of a row of their national flags at the Metropole before heading to dinner together.
Mr Trump told reporters he thought the talks would be very successful, and when asked if he was "walking back" on denuclearisation demands, said "no".
At their historic first summit in Singapore last June, Mr Trump and Mr Kim pledged to work toward denuclearisation and permanent peace on the Korean peninsula but little progress has been made.
Mr Kim said they had overcome obstacles to hold the second summit and praised Mr 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," Mr Kim said.
Mr Trump and Mr Kim held a 20-minute, one-on-one chat before sitting down to dinner with US Secretary of State Mike Pompeo, Mr Trump's acting Chief of Staff Mick Mulvaney, Mr 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 at the Metropole, beginning with another one-on-one session lasting 45 minutes, the White House said.
It said they would hold a "joint agreement signing ceremony," at the end of their meetings, followed by a news conference by Mr Trump.
The White House has given no indication as to what the signing ceremony might involve, although the two sides have held discussions that have included the possibility of a political statement to declare the 1950-53 Korean War over. North and South Korea have been technically at war since the conflict, with the Americans backing the South, ended in a truce, not a treaty.
Asked if he would declare a formal end to the Korean War, which North Korea has long called for, but some analysts say would be premature, Mr Trump said: "We'll see."
"We're going to have a very busy day tomorrow, " a smiling, relaxed-looking Mr Trump said before dinner, seated beside Mr Kim at a round table with the other four officials and two interpreters. "Our relationship is a very special relationship."
Experts said the pair were at pains to show their relationship had improved since their first meeting, with their body language closely mirroring each other.
Mr Trump and Mr Kim's Singapore summit, the first meeting between a sitting US 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, Mr 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.
He has also said he would be happy as long as North Korea, which has not tested a nuclear weapon or intercontinental ballistic missile since 2017, maintained that freeze.
US 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. Pyongyang will not commit to significant disarmament unless punishing US-led economic sanctions are eased, analysts said.
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.