Trump, speaking after a landmark summit with Kim on June 12 in Singapore, said that both leaders were "prepared to start a new history and write a new chapter between our nations".
Both leaders expressed optimism throughout roughly five hours of talks, with Trump thanking Kim afterward "for taking the first bold step toward a bright new future for his people".
Kim, who barely spoke until a signing ceremony at the end of the summit despite the presence of his translator, did not respond.
The United States and North Korea have interpreted the term "denuclearization" in different ways in the past, with Washington's definition referring to ridding Pyongyang of its nuclear arsenal and North Korea framing it as an end to the perceived USA ability to strike it with nuclear weapons from afar and to the protective U.S.
Language on North Korea's bombs was similar to what the leaders of North and South Korea came up with at their own summit in April. Trump called the drills "provocative and inappropriate", adding that he ultimately wants to "bring our soldiers home" - a reference to some 25,000 US troops stationed in South Korea.
"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 risky problem for the world", Trump said.
"Yeah, they are going to stay".
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.
Trump responded to such commentary on Twitter, saying: "The fact that I am having a meeting is a major loss for the US, say the haters & losers".
The United States and the DPRK will join their efforts to build a lasting and stable peace regime on the Korean peninsula.
"Our whole relationship with North Korea and the Korean Peninsula is going to a very much different situation than it has in the past", he said.
President Trump shook hands with Kim Jong Un at the start of a first-of-its-kind meeting of a sitting US President and a North Korean leader.
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Reporters pressed the president on the issue during the news conference. "And it will. It will soon end".
The communication in North Korea is quite known, mostly for its handful of channels.
Aware that the eyes of the world were on a moment many people never expected to see, Kim said many of those watching would think it was a scene from a "science fiction movie".
"I was pretty excited to see them", said the Stomper.
Trump, beaming over what seemed nearly a bromance with one of the world's most brutal dictators, said he would invite Kim to the White House.
Critics of the summit leapt at the leaders' handshake and the moonlight stroll Kim took Monday night along the glittering Singapore waterfront, saying it was further evidence that Trump was helping legitimise Kim on the world stage as an equal of the U.S. president.
Rodman later fought back tears after he talked of returning to the United States after a visit to North Korea and then getting "so many death threats" for trying to engage in a kind of soft diplomacy.
And the president reiterated that North Korea had committed to denuclearization.
The White House said that Trump would leave Singapore earlier that expected - on Tuesday night local time. Pompeo held firm to Trump's position that sanctions will remain in place until North Korea denuclearizes - and said they would even increase if diplomatic discussions did not progress positively.
"He called Kim "very smart" and a "very worthy, very hard negotiator". "Glad I fired him on Apprentice!"
After the two leaders met, they had another bilateral meeting with their staff that lasted about an hour and a half before they proceeded to lunch together at the luxury resort on Singapore's Sentosa Island.
Trump said at a post-summit news conference that he has held off from imposing additional sanctions, but that the U.S. would remove sanctions when the North's nuclear weapons "are no longer a factor".
He would not say whether that included the possibility of withdrawing USA troops from the Korean Peninsula, but said the U.S. was "prepared to take what will be security assurances that are different, unique, than America's been willing to provide previously".