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.
Trump said he expected the denuclearisation process to start "very, very quickly". US Secretary of State Mike Pompeo and North Korean officials would hold follow-up negotiations "at the earliest possible date", the statement said.
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.
‘Leaving past behind’
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.
"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.
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. — REUTERS/VNS