Update 10.20am: The two Koreas have agreed to rid their peninsula of nuclear weapons in historic talks between the two leaders.
A joint statement issued by Kim Jong Un and Moon Jae-in after the summit said the two had confirmed their goal of achieving "a nuclear-free Korean peninsula through complete denuclearisation".
The statement did not provide any new specific measures how to achieve the objective.
North Korea has placed its nuclear weapons up for negotiations. It has previously used the term "denuclearisation" to say it can disarm only when the US withdraws its 28,500 troops in South Korea.
The two countries also said they will jointly push for talks with the US, and potentially China, to officially end the 1950-53 Korean War, which stopped with an armistice and left the Koreas still technically at war.
North Korean leader Mr Kim and South Korean President Moon announced after their summit that the Koreas will push for three-way talks including Washington or four-way talks that also include Beijing on converting the armistice into a peace treaty and establishing permanent peace on the Korean peninsula.
The Koreas said they hope the parties will be able to declare an official end to the war by the end of this year.
While President Donald Trump has given his "blessing" for the Koreas to discuss an end to the war, there can be no real solution without the involvement of Washington and other parties that fought in the war because South Korea was not a direct signatory to the armistice that stopped the fighting.
"I feel like I'm firing a flare at the starting line in the moment of (the two Koreas) writing a new history in North-South relations, peace and prosperity," Mr Kim told Mr Moon as they sat at a table, built so that exactly 2018 millimetres separated them, to begin their closed-door talks.
Mr Moon responded that there were high expectations that they produce an agreement that will be a "big gift to the entire Korean nation and every peace-loving person in the world".
Mr Kim acknowledged widespread scepticism over their summit: "We have reached big agreements before but were unable to fulfil them. There are sceptical views on whether the meeting today will yield meaningful results.
The pair vowed to have more meetings, according to Mr Moon's spokesman, Yoon Young-chan, with Mr Kim joking that he would make sure not to interrupt Mr Moon's sleep any more, a reference to the North's drumbeat of early-morning missile tests last year.
Mr Kim also referred to a South Korean island the North attacked with artillery in 2010, killing four, saying the residents of Yeonpyeong Island have high hopes the summit will help heal past scars.
12 security guards jog alongside Kim Jong Un's vehicle as he is driven away from the first session of the summit with South Korean Pres. Moon Jae-in. https://t.co/KknLAPkQkW pic.twitter.com/KVaCgTziog— ABC News (@ABC) April 27, 2018
Mr Kim said he would visit Seoul's presidential Blue House if invited.
Earlier, both leaders smiled broadly as Mr Moon grasped Mr Kim's hand and led him along a red carpet into South Korean territory, where schoolchildren gave Mr Kim flowers and an honour guard stood at attention for inspection, a military band playing traditional Korean folk songs.
Earlier: Kim Jong Un makes history as North Korea leader crosses border to meet rival
With a single step over a weathered, cracked slab of concrete, North Korean leader Kim Jong Un made history by crossing over the world's most heavily armed border to greet his rival, South Korean president Moon Jae-in.
Mr Kim then invited Moon to cross briefly north with him before they returned to the southern side for talks on North Korea's nuclear weapons.
Those small steps must be seen in the context of the last year - when the United States, its ally South Korea and the North seemed at times to be on the verge of nuclear war as the North unleashed a torrent of weapons tests.
The moment must also be viewed in light of the long, destructive history of the rival Koreas, who fought one of the 20th century's bloodiest conflicts and even today occupy a divided peninsula that is still technically in a state of war.
"I feel like I'm firing a flare at the starting line in the moment of (the two Koreas) writing a new history in North-South relations, peace and prosperity," Mr Kim told Mr Moon as they sat at a table, its precise dimension of 2018 millimetres separating them, to begin their closed-door talks.
Mr Moon responded that there were high expectations that they produce an agreement that will be a "big gift to the entire Korean nation and every peace loving person in the world".
It was all smiles on Friday as Mr Moon grasped Mr Kim's hand and led him along an blindingly red carpet into South Korean territory, where school children placed flowers around their necks and an honour guard stood at attention for inspection.
Beyond the surface, however, it is still not clear whether the leaders can make any progress in closed-door talks on the nuclear issue, which has bedevilled US and South Korean officials for decades.
North Korea's nuclear and missile tests last year likely put it on the threshold of becoming a legitimate nuclear power. North Korea claims it has already risen to that level.
Mr Kim's news agency said that the leader would "open-heartedly" discuss with Mr Moon "all the issues arising in improving inter-Korean relations and achieving peace, prosperity and reunification of the Korean peninsula" in a "historic" summit.
It is the first time one of the ruling Kim leaders has crossed over to the southern side of the Demilitarised Zone (DMZ) since the Korean War ended in 1953.
The greeting of the two leaders was planned to the last detail.
Thousands of journalists were kept in a huge conference centre well away from the summit, except for a small group of tightly-controlled pool reporters at the border.
Mr Moon stood near the Koreas' dividing line, moving forward the moment he glimpsed Mr Kim appearing in front of a building on the northern side.
They shook hands with the border line between them.
Mr Moon then invited Mr Kim to cross into the South; Mr Kim invited Mr Moon into the North, and they then took a ceremonial photo facing the North and then another photo facing the South.
Two fifth-grade students from the Daesongdong Elementary School, the only South Korean school within the DMZ, greeted the leaders and gave them flowers.
Mr Kim and Mr Moon then saluted an honour guard and military band, and Mr Moon introduced Mr Kim to South Korean government officials.
Mr Kim returned the favour with the North Korean officials accompanying him.
They posed for a photo inside the Peace House, where the summit was to take place, in front of a painting of South Korea's Bukhan Mountain, which towers over the South Korean Blue House presidential mansion.
Nuclear weapons will top the agenda, and Friday's summit will be the clearest sign yet of whether it is possible to peacefully negotiate those weapons away from a country that has spent decades doggedly building its bombs despite crippling sanctions and near-constant international opprobrium.
Expectations are generally low, given that past so-called breakthroughs on North Korea's weapons have collapsed amid acrimonious charges of cheating and bad faith.
Sceptics of engagement have long said that the North often turns to interminable rounds of diplomacy meant to ease the pain of sanctions - giving it time to perfect its weapons and win aid for unfulfilled nuclear promises.
Advocates of engagement say the only way to get a deal is to do what the Koreas will try on Friday: Sit down and see what is possible.
Mr Moon, a liberal whose election last year ended a decade of conservative rule in Seoul, will be looking to make some headway on the North's nuclear programme in advance of a planned summit in several weeks between Mr Kim and US president Donald Trump.
Mr Kim, the third member of his family to rule his nation with absolute power, is eager, both in this meeting and in the Trump talks, to talk about the nearly 30,000 heavily armed US troops stationed in South Korea and the lack of a formal peace treaty ending the Korea War - two factors, the North says, that make nuclear weapons necessary.
North Korea may also be looking to use whatever happens in the talks with Mr Moon to set up the Trump summit, which it may see as a way to legitimise its declared status as a nuclear power.
One possible outcome on Friday, aside from a rise in general goodwill between the countries, could be a proposal for a North Korean freeze of its weapons ahead of later denuclearisation.
Seoul and Washington will be pushing for any freeze to be accompanied by rigorous and unfettered outside inspections of the North's nuclear facilities, since past deals have crumbled because of North Korea's unwillingness to open up to snooping foreigners.
South Korea, in announcing Thursday some details of the leaders' meeting, acknowledged that the most difficult sticking point between the Koreas has been North Korea's level of denuclearisation commitment.
Mr Kim has reportedly said that he would not need nuclear weapons if his government's security could be guaranteed and external threats were removed.
Whatever the Koreas announce on Friday, the spectacle of Mr Kim being feted on South Korean soil will be something to behold.