Moon recently said the two Koreas should formally agree to stop the war and sign a peace treaty, days after Trump gave his "blessing" for them to discuss an end of the war.
KCNA news agency also said Kim would plant a memorial tree with Moon at the meeting as a symbol of growing peace between the nations. The meeting could take place as early as June, the president has said.
In a surreal moment for Korea watchers, Kim stepped across the military demarcation line into South Korea and shook his southern counterpart's hand - something that would have been nearly unthinkable just a few months ago.
Tension over North Korea's tests of nuclear weapons and ballistic missile surged previous year and raised fears of USA military action, after North Korean threatened to develop a nuclear weapon capable of hitting the United States.
TRT World's Oliver Whitfield-Miocic has more.
After being led along a red carpet by South Korean honour guard in traditional blue, yellow and red outfits, the two men entered the Peace House on the South side.
Kim repeatedly expressed hopes for peace between the two Koreas, which remain technically at war since the 1950-1953 Korean War ended in a cease-fire.
Official talks between Mr Moon and Mr Kim will begin at 9.30am local time at the Peace House in Panmunjom.
"A new history starts now".
And if "peace" is formally declared as part of a wider deal with the Americans, the North can make a new case for its long-cherished goal: to have the U.S. withdraw its large contingent of troops from the South.
Shaking hands again, Moon, 65, and Kim, 34, turned to face photographers on the North and then the South before Kim grabbed Moon's hand and, in an unplanned move, invited him to step across the border into the North, where they stood face-to-face to talk a bit more. Kim said that if Moon comes, he will make Moon conveniently visit the DPRK mountain.
Kim Jong-un's message that he wrote in the Peace House visitor's book.
"First is a commitment to improve inter-Korean relations".
The North Korean nuclear threat was a top issue of discussion during the prime minister's recent meetings with various European leaders.
South Korea and the United States will wrap up their annual Foal Eagle joint military drills involving tens of thousands of troops ahead of the summit, Seoul's defence ministry said Thursday, and suspend the Key Resolve command post exercise for the day.
Not only is the physical meeting of the two Korean leaders historic, merely engaging in a conversation is as well.
Thousands of journalists were kept in a huge conference center well away from the summit, except for a small group of tightly-controlled pool reporters at the border. "And after all, he (Kim) can settle claims against NK in any US-NK agreement", Baker said by email. They then returned to the South Korean side together.
The North's nuclear arsenal will be high on the agenda.
Dr Petrov, a visiting fellow at the Australian National University College of Asia and the Pacific, said Kim desperately needs to steer North Korea away from an imminent disaster such a nuclear war, a domestic upheaval or both.
Also in the delegation are Ri Su Yong, a member of the North's politburo and Ri Myong Su, the chief of the general staff of the Korean People's Army.
"The establishment of permanent peace on the Korean Peninsula".
There has been speculation that a meeting between Kim and U.S. President Donald Trump in the coming weeks could possibly be held in Southeast Asia.
North Korea may also be looking to use whatever happens in the talks with Moon to set up the Trump summit, which it may see as a way to legitimise its declared status as a nuclear power.