This media briefing came from Yoon Young-chan, a senior public communications secretary for Moon Jae-in. He's given us a comprehensive South Korean take on the talks so far.

Key points:

• That unexpected step by Mr Moon into North Korea was not planned in advance (this is becoming a talking point on Korean social media).
• Mr Kim invited Mr Moon to North Korea, and said he would receive the same welcome he did.
• Mr Kim said his short walk over the border "took more than 10 years" but they should meet more often from now on.
• "Let's never go back to starting point. Let's create a better world - I will do my best" - Mr Kim to Mr Moon.
• Moon Jae-in said the Korean people should be able to take the lead for other countries to follow.

Image source, EPA

Mr Kim, we're told, is concerned that Mr Moon might encounter bad traffic conditions if he travels to the North. When relations improve, he says, the two sides can join up their roads.

Also on roads, Mr Moon expressed his sadness over a recent bus crash in North Korea, which killed 32 Chinese tourists. That prompted some rare media coverage in the North.

South Korea officials are giving the media a rundown on what happened this morning.

Like all good visits, the conversation started with the leaders sharing details of their journey this morning and how early they had to get up.

Mr Kim, apparently, told Mr Moon he would make sure he wasn't woken up so early in the future by North Korean missile tests.

Image source, AFP

Hello, if you're just joining our live coverage of a hugely significant day on the Korean peninsula. The Korean leaders are taking a lunch break, so there's time for a recap.

• North Korea's Kim Jong-un has made history by stepping onto South Korea territory
• He was met at the border by South Korea's Moon Jae-in, with smiles and many handshakes
• In an unexpected move, Mr Moon also briefly stepped over into North Korea for a handshake
• After a ceremonial welcome, they had a private meeting, then a televised conversation.
• Both leaders - technically still at war - welcomed the developments and said they wanted to move their relationship forward.
• We don't yet know what will be agreed today, nor how lasting an impact it will have.
• Read more: What to to expect from the talks

The leaders have been in their private chats for over an hour now. What have they been talking about? The world is dying to know. They're due a break for lunch soon - which will be eaten separately on their own sides. Fear not - barring major incidents, they'll be eating dinner together later on.

Image source, AFP

Here's something unexpected. Gambling site Ladbrokes is currently offering odds of 2-1, external on Donald Trump and Kim Jong-un winning the Nobel Peace Prize. (Moon Jae-in may have something to say about that.)

Mr Trump certainly thinks he deserves the credit, external, and our colleagues at BBC Reality Check have been checking out how right that view is.

Confused by the blow-by-blow coverage? Take a step back and catch up on the North Korea crisis in a nutshell. Well, in 300 words., external

Interesting insight from AFP reporter Sunghee Hwang.

Pictures of South Koreans watching news of North Korea on TV are usually the only images we have available to illustrate such stories. Today is the exception as it's all been televised live and the world's media are there.

But for nostalgia's sake, here are some South Koreans watching events on big screens.

Image source, EPA

Image source, EPA

Image source, Getty Images

Image source, Reuters

Image source, Getty Images

In case you missed it, here's the moment Mr Moon and Mr Kim shook hands at the border.

Want more fuel for speculation on how the summit will go? OK let's look at South Korean shares. Seoul's Kospi index is moving about 1% higher and it's been on an upward trend for the past days.

And yes, most pundits are pinning this on optimism around the summit.

North Korean stocks meanwhile... OK just kidding. Pyonyang does not have a stock exchange, there's no place for that in old school communism.

Image source, AFP

At both the first two inter-Korean summits, the wives of the two sides' leaders were nowhere to be seen.

But North Korea appears to be moving Kim Jong-un's wife Ri Sol-ju into a more public position, and there has been speculation she might be part of the summit this time.

We haven't seen her so far but some reports suggest the first lady of the North might turn up for the big dinner later.

The US/NK summit is still far from certain. The date, agenda and venue are all mere speculation. So whether or not the next handshake will be between Mr Trump and Mr Kim might well depend on the outcome of today.

The BBC's Laura Bicker in Seoul wrote this comprehensive piece as news of these talks emerged a few weeks ago on how talking to Kim could be, for both Trump and Moon, the political gamble of the century.

Image source, EPA

One goal of those historic talks will be to prepare for the even more historic (yes!) talks between Donald Trump and Kim Jong-un.

Just hours before today's meeting got under way, the White House chose its moment to release pictures of Mike Pompeo - then-CIA chief and now secretary of state - meeting the North Korean leader several weeks ago.