The two countries plan to sign a peace treaty, formally ending the Korean War.

"There will be no more war on the Korean peninsula and thus a new era of peace has begun," the statement says.

The two leaders agreed to work for the "complete denuclearisation of the Korean peninsula".

The leaders of North and South Korea have just signed a joint declaration.

What is Byeongjin, and where is Mount Kumgang?

Our colleagues at BBC Monitoring have put together an explainer on the key phrases observers are listening for.

How often does Kim Jong-Un speak unscripted? The former deputy commander of the South Korean army, Lt-Gen In-Bum Chun, had this to tell BBC Radio 4's Today programme.

The US says it hopes the summit will "achieve progress towards a future of peace and prosperity".

It wishes the Korean people well and looks forward to "robust discussions".

The next big summit, of course, is the one planned between US President Donald Trump and Kim Jong-un in the coming months.

Plenty of observers have mentioned the peaceful sound of birds in the background as the two leaders spoke privately.

That's because, as journalist Kim Gamel points out, the Demilitarised Zone along the border has become something of a wildlife refuge.

In 2013, National Geographic published a photo gallery of the area, writing, external: "Species that have dwindled or disappeared in some parts of Asia have found refuge in the DMZ."

If you're looking for a (very) concise summary of the factors that drove the two leaders to his summit, here it is in 300 words.

North Korea state television has referred to Moon Jae-in as “president”, BBC Monitoring reports. Traditionally, they don’t use the title as they don’t consider South Korea to be a separate country, calling it south Korea (with a small s) or “the southern part of Korea”.

Much of the talking seems to have taken place during the private walk after the tree ceremony.

Mr Kim and Mr Moon conferred alone - and out of range of the microphones.

When they sat on a blue footbridge, only the chirping of birds in the trees could be picked up.

This is the first summit between the two states in over a decade. Only two others have been held since the end of the Korean War in 1953 - both in Pyongyang, in 2000 and 2007.

After the 2000 meeting, the then South Korean President, Kim Dae-jung, won the Nobel Peace Prize for his "sunshine policy" of rapprochement with the North.

Image source, AFP

In 2007, it was Roh Moo-hyun's turn to meet Kim Il-sung. They agreed steps on denuclearisation but since then the North has carried out six nuclear tests and says it has developed a nuclear bomb small enough to be carried by a long-range missile.

Mr Moon was elected president of South Korea in 2017 after a corruption scandal drove his predecessor, Park Geun-hyeout, out of office.

Kim Jong-un took over the reins of power in the North after his father Kim Il-sung's death in 2011.

Image source, Getty Images

Mr Moon's wife Kim Jung-sook will also join her husband at tonight's dinner.

Just like her North Korean counterpart, she is a former professional singer and the 63-year-old has put those skills on display before, performing pop star Psy's horse-riding Gangnam Style dance on a visit to the Philippines last year.

She's won fans in South Korea with her outgoing personality, and its contrast to her husband who is often described as quiet...and even bland.

We didn't think the topic of defectors from North Korea would come up during the morning's chat - Kim Jong-un almost never talks about them.

But according to NK News, he talked to Mr Moon about seeing "displaced people, defectors and residents of the island of Yeonpyeong" - a South Korean island by the border - en route, noting they had "high expectations" of the visit.

“We should value this opportunity so that the wound between the two Koreas can be healed," he said. "The borderline is not high, it will be eventually erased if a large number of people pass over."

After the tree-planting ceremony, Mr Kim and Mr Moon took a brief walk.

Image source, Reuters

At the tree-planting ceremony, Mr Kim's sister Kim Yo-jong helped him don white gloves, as he and Mr Moon laid the final shovelful of earth.

Mr Kim said a "new spring" had come for the two countries.

After they planted the tree, Mr Kim told Mr Moon: "Just like a pine tree, I hope that we can always be green, even in winter time."

"Yes, it will be like that," Mr Moon replied.

Image source, Inter-Korean Summit Press Corps

This is a picture that paints a thousand words - officials spraying disinfectant around a table in Peace House just before Kim Jong-un sat down to sign the visitor's book. Reports do occasionally emerge saying the North Korean leader is paranoid about germs and being poisoned. They're never that well sourced but this image does reflect the level of preparation that's gone into today's summit.

The two leaders have just planted that tree, in a ceremony laden with symbolism.

It was planted on the military demarcation line - the formal border between the two countries.

The tree itself dates from 1953, the year the Korean war armistice was signed, and the soil and water used in its planting were taken from both sides of the border and mixed.

Image source, AFP

One face at tonight's dinner that's sure to attract plenty of attention is North Korea's first lady, Ri Sol Ju.

The former professional singer has gained a growing political profile in recent months.

She's been seen by Mr Kim's side at big events at home and abroad, including last month's summit in China. That's a big shift from the past, when she was relegated to the background and simply referred to as Mr Kim's wife.

So will she take centre stage tonight?

If you're just joining, welcome to our live coverage of events on the Korean border, where history is being made. Here's a catch-up:

• Kim Jong-un has become the first North Korean leader to set foot in South Korea since the end of Korean War
• He was met by South Korea's Moon Jae-in, all smiles and with a very symbolic handshake
• In an unexpected move, Mr Moon also briefly stepped over into North Korea for another handshake
• A briefing earlier in the day was a rare chance to hear Mr Kim speaking in a live broadcast

• Mr Kim reportedly told Mr Moon he was there "to put an end to the history of conflict"
• It is mid-afternoon local time. The morning session of talks is over, and the two leaders are scheduled to take part in a joint tree-planting ceremony before the second round of talks

Read our main story: Kim Jong-un crosses into South Korea