The two leaders capped their summit on Thursday with a visit to Mount Paektu, the mythological birthplace of the Korean people and one of the most powerful symbols in North Korean propaganda.
Talks between the USA and North Korea have stalled in recent weeks after President Trump became uncertain that Kim was following through with pledges made at their Singapore summit in June.
He also says the Trump administration had invited North Korean officials to meet in Austria with the new US special envoy for North Korea "at the earliest opportunity". Compared to the vague language of their two earlier summits, Mr Kim and Mr Moon seem to have agreed on an ambitious program meant to tackle soaring tensions that caused many to fear war a year ago as the North tested a string of increasingly powerful weapons.
On Thursday, Moon hinted he is aware of Kim's demands of Trump, but he refused to give details saying the matter should be discussed between the USA and North Korean leaders. In return, the North said it was seeking "corresponding measures" from the U.S.
Kim Jong-un, his North Korean counterpart, watched from a distance.More news: Here's How Xiaomi Mocked Apple For Steep Pricing Of New iPhones
"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.
Previously, Pyongyang has asked for "phased and synchronised measures" for denuclearisation, meaning the global community would slowly relax or drop sanctions with each step in the denuclearisation process, rather than waiting until its completion.
Diplomacy between the US and North Korea that had gotten nowhere since the summit clicked into gear again.
"There is nothing the North has offered so far that would constitute irreversible movement toward denuclearisation, however you define that, by January 2021 or any other time, or even a reduction of the military threat it poses to the South and the region", said a USA intelligence official. It is a very meaningful outcome.
On Wednesday, Trump tweeted about North Korea's commitment to denuclearization, saying Kim has agreed to "nuclear inspections" and to "permanently dismantle a test site and launch pad".
Kim first proposed a second meeting with Trump after their unprecedented June summit in Singapore.
Washington calls for concrete action, such as a full disclosure of North Korea's nuclear and missile facilities, before satisfying Pyongyang's key demands, including an official end to the war and the easing of worldwide sanctions.More news: Trump Visits Hurricane-Ravaged North Carolina
Many also believe the North has other covert nuclear facilities.
An end-of-war declaration would not affect the presence of US troops and the UNC in the South, Moon said, adding that Kim shared his view.
Ahead of this week's talks, it was expected that two leaders would continue to work to formally end to the Korean War, which ended in an armistice 65 years ago. "This could very well set the conditions for the president to withdraw US forces (from South Korea) as he has stated is his desire", Maxwell said. Those factors combine to put time on Moon's side: North Korea already has nuclear weapons but nearly certainly will not use them unless attacked.
"The biggest outstanding question still remains - what price the North Koreans want to extract from the United States", Fuchs said.
The two leaders also agreed to pursue a bid to co-host the 2032 Olympic Games, and Moon said he expects Kim Jong-un to soon become the first North Korean leader to ever visit Seoul.
"Chairman Kim Jong Un has again and again affirmed his commitment to denuclearization", Moon said after returning to Seoul.
Their main goal, he added, was "to appease Trump, so that Moon and Kim can keep their engagement alive".More news: Tues. 12:40 pm: Trump declassifies some Russia probe papers