- Kim Jong Un said meeting "a good prelude to peace"
- No immediate word on any substantive progress in discussions
- Donald Trump said he forged a "good relationship" with Kim
But a joint statement signed at the end of their historic summit in Singapore gave few details on how either goal would be achieved.
"President Trump committed to provide security guarantees to the DPRK and Chairman Kim Jong Un reaffirmed his firm and unwavering commitment to complete denuclearisation of the Korean Peninsula," said the statement.
DPRK is the Democratic People's Republic of Korea, the formal name of North Korea.
Trump said he expected the denuclearisation process to start "very, very quickly". U.S. Secretary of State Mike Pompeo and North Korean officials would hold follow-up negotiations "at the earliest possible date", the statement said.
"It is unclear if further negotiations will lead to the end goal of denuclearisation," said Anthony Ruggiero, senior fellow of Washington's Foundation for Defense of Democracies think tank. "This looks like a restatement of where we left negotiations more than 10 years ago and not a major step forward."
The document also made no mention of the international sanctions that have crippled North Korea's economy for pursuing its nuclear weapons programme.
Nor was there any reference to finally signing a peace treaty. North Korea and the United States were on opposite sides in the 1950-53 Korean War and are technically still at war, as the conflict, in which millions of people died, was concluded only with a truce.
But the joint statement did say the two sides had agreed to recovering the remains of prisoners of war and of those missing in action and repatriating them.
"At the same time, there needs to be a peace mechanism for the peninsula to resolve North Korea's reasonable security concerns," China's top diplomat, State Councillor Wang Yi, told reporters in Beijing.
If the joint statement does lead to a lasting detente, it could fundamentally change the security landscape of Northeast Asia, just as former U.S. President Richard Nixon visit to Beijing in 1972 led to the transformation of China.
But Li Nan, senior researcher at Pangoal, a Beijing-based Chinese public policy think tank, said the meeting had only symbolic significance.
"Leaving past behind"
However, the dollar jumped to a 3-week top on Tuesday and Asian shares rose on news of the agreement.
Before signing what Trump described as a "comprehensive" document, Kim said the two leaders had a historic meeting "and decided to leave the past behind. The world will see a major change."
"People are going to be very impressed and people are going to be very happy and we are going to take care of a very dangerous problem for the world," Trump said.
He called Kim "very smart" and a "very worthy, very hard negotiator."
"I learned he's a very talented man. I also learned that he loves his country very much."
During a post-lunch stroll through the gardens of the Singapore hotel where the summit was held, Trump said the meeting had gone "better than anybody could have expected".
Kim stood silently alongside, but the North Korean leader had earlier described their summit as a "a good prelude to peace".
They had appeared cautious and serious when they first arrived for the summit at the Capella hotel on Singapore's Sentosa, a resort island with luxury hotels, a casino, manmade beaches and a Universal Studios theme park.
But, with cameras of the world's press trained on them, they displayed an initial atmosphere of bonhomie as they met on the verandah of the Capella, a refurbished 19th century British regimental officers' mess.
Body language expert said both men tried to project command as they met, but also displayed signs of nerves.
"Like a movie"
After a handshake, they were soon smiling and holding each other by the arm, before Trump guided Kim to the library where they held a meeting with only their interpreters. Trump had said on Saturday he would know within a minute of meeting Kim whether he would reach a deal.
Inside, they sat alongside each other against a backdrop of North Korean and U.S. flags, with Kim beaming broadly as the U.S. president gave him a thumbs up.
After initial exchanges lasting around 40 minutes, Trump and Kim emerged, walking side-by-side through the colonnaded hotel before entering a meeting room, where they were joined by their most senior officials.
Kim was heard telling Trump through a translator: "I think the entire world is watching this moment. Many people in the world will think of this as a scene from a fantasy...science fiction movie."
As the two leaders met, Singapore navy vessels, and air force Apache helicopters patrolled, while fighter jets and an Gulfstream 550 early warning aircraft circled.