Mr Trump and Mr Kim - who traded vicious insults a year ago - gripped each other's hands in front of U.S. and North Korean flags.
"I really believe otherwise I wouldn't be doing this", Trump said of the summit.
Trump said Tuesday the summit has helped bring the world back from "nuclear catastrophe".
The Australian government has cast doubt over whether Donald Trump has suspended military exercises with South Korea, noting that was not part of the joint declaration between the United States and North Korea. The omission of the words "verifiable" and "irreversible" from the phrasing on denuclearization suggested North Korean resistance to Trump's requests. "The V matters", he told reporters in Singapore.
The analyst, Moon, advised the North's report should be confirmed from the USA side, while another expert agreed the North Korean state media should be taken with a grain of salt.
It said that Kim and Trump had "shared recognition to the effect that it is important to abide by the principle of step-by-step and simultaneous action in achieving peace, stability and denuclearization of the Korean Peninsula". Trump, it added, had expressed his "understanding" of the stance and had conveyed "his intention to halt" the exercises.
The drills - often called "war games" - are held in South Korea with local forces and U.S. soldiers stationed there.
Korean paper quick to carry many photos of Kim's Singapore trip
The report added that more meetings could come on their respective home turf.
If, as Trump said on Tuesday he sensed, Kim wants the grand bargain of disarmament for economic integration, the personal endorsement from the two leaders on negotiations may be all that counts. Kim asked Trump to travel to Pyongyang "at a convenient time", the news agency said. Now, with Donald Trump's summit with Kim Jong-un, the tables have been turned.
The KCNA report noted Kim's "proactive peace-loving measures" and the "will of the top leaders of the two countries to put an end to the extreme hostile relations" that date back to the Korean War almost 70 years ago. "The U.S. has yielded major concessions - the summit itself and the end of our defensive military exercises - in return for what appears to be a vague and undefined premise from the North Korean leader".
Mr Trump's announcement, made in a press conference at the end of the historic summit with Mr Kim, could overturn longstanding U.S. policy and was described by experts as an "astonishing" concession. "Thank you to Chairman Kim, our day together was historic!"
Rodong Shinmun devoted its entire front page to the summit.
As Trump and Kim approached one another from opposite wings of a makeshift stage, complete with a red carpet and a row of alternating American and North Korean flags, Kim had established himself as an equal, for one day at least, to the most powerful leader on Earth.
Trump had been scheduled to fly back to Washington tomorrow morning after spending today with Kim in Singapore, but on the eve of the summit, he altered his schedule, opting to return at about 8pm today after a full day of meetings with Kim - nearly 15 hours earlier than previously anticipated.
The 24-year-old USA singer has been replying to fans tweeting her about rumours that she is to marry boyfriend, Pete Davidson . When one fan said "I hope he knows he is marrying us as well", Grande replied with "HAHAHAHAHAHH HES BEEN BRIEFED" (sic).
We will file an objection against the decision issued by the KBA (German Federal Motor Transport Authority). This comes just three years after VW admitted having fitted "cheat" devices in millions of its cars.
Navarro added that while these were his own words, they reflected the "the sentiment that was on Air Force One". Trump's latest twitter tirade came after he arrived in Singapore for his summit with North Korea's Kim Jong-un.
The two clasped hands for a long while Tuesday as they posed for photos in front of a row of USA and North Korean flags. Perhaps most importantly, Trump did not succeed in getting Kim to clarify his definition of "denuclearization".
The Metro 2033 and Metro : Last Light games are based on Dmitry Glukhovsky's Metro 2033 and Metro 2035 novels. He goes on an epic journey that spans the continent and goes across post-apocalyptic Russian Federation .
First, "MPs voted to reverse the Lords amendment removing the "exit day" from the bill by 326 votes to 301 - a majority of 25". They subsequently warned they will not be easily bought off, while Tory Brexiteers urged ministers not to concede any ground.
Canadian Foreign Minister Chrystia Freeland said her nation "does not conduct its diplomacy through ad hominem attacks". Tensions between Trump and Trudeau are at an all-time high after the two leaders disagreed over trade at the summit.
The Academy Award victor shouted out, " Fuck Trump! ", a declaration which brought the audience to their feet in ovation. He made it through University of Pennsylvania, so he was exposed to a quality education, but he's still an idiot'.
Justify accomplished a lot in a very short time. "But if you win the Triple Crown you become a legend and that is what he is now". When Geroux broke next to last I would have thought that was a great scenario, he doesn't have to worry about him.
Antoine Griezmann has made a decision on his future
However, at a France press conference on Tuesday afternoon, he declined to give an update on his future intentions. La Liga president Javier Tebas has declared he expects Antoine Griezmann to stay put this summer.
Government forced to climbdown to avoid Brexit vote defeat
Shortly before 1.30pm on Tuesday, David Davis, the Brexit Secretary, opposed a House of Lords amendment in no uncertain terms. In his letter, Mr Davis said it was "simply not right" that Parliament could effectively overturn the referendum result.
Alec Baldwin: I Could 'Absolutely' Beat Trump in 2020
His impersonation - which won the actor an Emmy award previous year - has drawn the ire of the president, especially on Twitter. Baldwin said he wants Democrats "to have a smart opposition to keep them on their toes".
Robbie Williams to perform at World Cup opening ceremony
The opening ceremony will take place at Moscow's Luzhniki Stadium. "Of course it is also an emotional one for the hosts". Brazil's two-time World Cup victor Ronaldo will also feature in a ceremony billed as consisting of a new format.
Judge OKs AT&T merger with Time Warner
Last fall, he presented AT&T and Time Warner with two options: Sell the majority stake in either DirecTV or Turner Broadcasting. In after-market trading following the ruling, shares of Time Warner rose almost 5 percent and AT&T shares fell 1.6 percent.
Justify wins Triple Crown with dominant victory at Belmont
In an email to Newsday, Hart wrote: "The best horse won. but Noble Indy got a raw deal in how the race was run". "I'm not. There is also, of course, the risk of defeat, which takes away some of the invincibility of a Triple Crown victor .
Paris hostages freed, suspect arrested
A pregnant woman also managed to escape and was being treated for shock by ambulance services on the scene. One hostage has a head injury, Yves Lefebvre, a police union spokesman, said.