- Donald Trump said both leaders were "prepared to start a new history"
- Talking to reporters, Trump said Kim had "an opportunity like no other"
- The US President also said he would be "stopping the war games"
Trump, speaking after his historic summit meeting with Kim in Singapore, said both leaders were "prepared to start a new history and write a new chapter between our nations".
"He reaffirmed his unwavering commitment to the complete denuclearisation of the Korean peninsula," Trump told a news conference.
Trump said Kim had "an opportunity like no other", and that Kim had said a North Korean missile testing site "is going to be destroyed very soon".
Trump also said he and Kim had discussed human rights briefly.
But a joint statement signed at the end of their historic summit in Singapore gave few details on how their goals 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.
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.
Political analysts said the summit had yielded only symbolic results and nothing tangible.
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.
China, the third party to the truce, said it hoped North Korea and the United States could reach a basic consensus on denuclearisation.
"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.
But Li Nan, senior researcher at Pangoal, a Beijing-based Chinese public policy think tank, said the meeting had only symbolic significance.
"It is too early to call it a turning point in North Korea-U.S. relations," Li said.
"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."
Trump said he had formed a "very special bond" with Kim and that relationship with North Korea would be very different.
Asked whether he would invite Kim to the White House, Trump said: "Absolutely, I will."
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".
Both men walked to Trump's bullet-proof limousine, nicknamed "The Beast", and looked in at the rear seat, with Trump apparently showing Kim something inside. They then resumed their walk.
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.
Trump was joined by Pompeo, National Security Adviser John Bolton, and John Kelly, White House Chief of Staff, for the expanded talks, while Kim's team included former military intelligence chief Kim Yong Chol, foreign minister Ri Yong Ho and Ri Su Yong, vice chairman of the ruling Workers' Party.
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.
After the meetings, the two teams and other senior officials met for a working lunch, where beef short ribs, sweet and sour pork and "Daegu Jormin", or Korean braised cod, were served for the main course, according to the menu. That was to be followed by dark chocolate tarts, pastries and vanilla ice cream for dessert. The North Korean leader's sister and close confidante Kim Yo Jong was among the lunch party.