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". U.S. Secretary of State Mike Pompeo and North Korean officials would hold follow-up negotiations "at the earliest possible date", the statement said.
Donald Trump was the first to arrive at the Capella Hotel on Sentosa, a resort island off Singapore's port. Television images showed Kim Jong Un's motorcade arriving later (Reuters)
The document also made no mention of the international sanctions that have crippled North Korea's economy for pursuing its nuclear weapons programme.
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.
US President Donald Trump holds up a document signed by him and North Korea's leader Kim Jong Un following a signing ceremony during their historic US-North Korea summit, at the Capella Hotel on Sentosa island in Singapore (AFP)
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.
US President Donald Trump and North Korean leader Kim Jong Un walk to shake hands at the Capella Hotel on Sentosa island in Singapore (Reuters)
"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.
An employee of a foreign exchange trading company in Tokyo looks at the screens displaying live news of meeting between Kim Jong Un and Donald Trump in Singapore (AFP)
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".
A view looking toward Sentosa, the resort island where Kim Jong Un and Donald Trump met today (Reuters)
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."
US President Donald Trump and North Korean leader Kim Jong Un walk after lunch at the Capella Hotel on Sentosa island in Singapore (Reuters)
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.
US President Donald Trump and North Korea's leader Kim Jong Un sign documents that acknowledge the progress of the talks and pledge to keep momentum going, after their summit at the Capella Hotel on Sentosa island in Singapore (Reuters)
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