Kim promised to accept global inspectors to monitor the closing of a key missile test site and launch pad and to visit Seoul soon, and both leaders vowed to work together to try to host the Summer Olympics in 2032.
Kim said on Wednesday he would visit Seoul in the near future, in what would be the first trip to the southern capital by a North Korean leader.
The United States is ready to restart negotiations with North Korea "immediately", US Secretary of State Mike Pompeo said on Wednesday (Sept 19), with the aim of completing North Korea's denuclearisation by January 2021.
Moon said Trump and Kim had both made pledges at their June summit in Singapore: North Korea promised to take steps toward complete denuclearization and returning the remains of US servicemen killed in the war, while the United States pledged to "end hostilities, guarantee the security of the regime, and establish a new relationship with North Korea", he said.
"[Moon] would add much legitimacy to Kim, who had never had any summit with a foreign leader until early this year", said Seo Yu-suk, a research manager at the Institute of North Korean Studies in Seoul.
Speaking at a news conference upon his return to Seoul on Thursday, Moon Jae-in said Kim told him during his three-day summit in Pyongyang that he wanted to hasten the process of denuclearisation. "We discussed some of it, but we did not have it written down so I can not reveal any of the discussed items with you here", he said. "I now propose that we completely eliminate the hostility of the past 70 years and take a big step forward in peace so that we can become one again".
"The first meeting between Kim Jong Un and Trump yielded a rather general agreement, but now Kim seems to have more concrete idea about what he wants to exchange with Trump", he said.
"State media and ideological indoctrination was setting the stage" for Kim Jong Il's power succession that took place in 1980, said Michael Madden, a North Korea leadership expert at the Stimson Centre's 38 North in Washington.
A stone marker commemorating South Korean President Moon Jae-in's visit to Pyongyang has the wrong dates carved on it. Kim say Mount Paektu has become a "subject of yearning" for South Koreans and Koreans living overseas.
Moon's office says North Korean workers made a mistake. "The need for US forces in Korea is not affected at all". North Korea has been demanding a declaration formally ending the Korean War, which was stopped in 1953 by a ceasefire, but neither leader mentioned it Wednesday as they read the joint statement.
Mr. Trump said they would wait and see what North Korea does next.