At their June 12 summit, Trump and Kim signed a declaration that stated, "President Trump committed to provide security guarantees to the DPRK, and Chairman Kim Jong Un reaffirmed his firm and unwavering commitment to complete denuclearization of the Korean Peninsula".
Moon Jae-in and Kim Jong Un agreed to turn the Korean peninsula into a "land of peace without nuclear weapons and nuclear threats" and take "prompt steps" toward the goal.
"We are not afraid of future challenges", he said. This could be a reference to Kim's insistence that Trump promised he would declare the Korean War to be officially over.
Washington wants North Korea to outline the entirety of its nuclear programme, and its response to Wednesday's joint statement from the Koreas remains to be seen. And soon after that, Kim was in Singapore with the president of archenemy the United States, Donald Trump.
The two countries also formed a unified women's ice hockey side - a move initially controversial in the South, with critics saying that Seoul was depriving its own citizens of a chance to play on the Olympic stage.
As part of the military agreement, the neighbors will halt border drills from November 1, Yonhap reports.
"We can take military measures to achieve the objective of the "armistice agreement" by making the demilitarized zone into peace zone for the first time in 65 years", the defense ministry said in a separate statement.
Do said that co-hosting the Games would build on the success of this year's Winter Olympics in Pyeongchang, where North Korea's participation helped ease tensions on the peninsula.
Other agreements aimed at removing some longstanding irritants from their relations - such as allowing more contact between families divided by the Korean War.
Unlike Trump's initial tweets praising the summit, the news brought a quick and negative response from Republican Sen. During prior negotiations in 2008, a cooling tower at the site was demolished, but North Korea's nuclear program continued.
North Korea's May Day Stadium is the largest in the world with a reported capacity of 150,000.
Harry Kazianis, director of defense studies at the Center for the National Interest in Washington, tells NPR that keeping the lines of communication open between North and South is paramount.
Mr Moon also said Mr Kim would try to visit Seoul sometime this year.
Kim has gone all out to make Moon's visit a memorable one.
The resumption of South Korean tours to the Mount Kumgang resort, meanwhile, suspended in 2008 after a South Korean woman was fatally shot by North Korean guards, could violate the United Nations sanctions banning the transfer of bulk cash to Pyongyang. Kim called the agreement "a leap forward" toward peace between the two historically bitter rivals.
Moon is expected to speak with Trump after this week's event, and Trump will then decide if he'll accept Kim's invitation.
The International Olympic Committee traditionally does not announce host cities until seven years ahead of the games. Almost 3,000 journalists watched her giving directions to South Korean President Moon Jae-in himself as the two Korean leaders stood on stage facing North Korean honor guards.