Mr Kim and Mr Trump are out of lunch and taking a stroll through the garden.

Mr Trump tells reporters there's going to be "a signing".

Here's a quick recap of Mr Kim and Mr Trump's comments to the press after their private chat.

Actually not that brief - it didn't last much longer than this.

Here's a glimpse into the working lunch between Mr Trump and Mr Kim, now taking place.

You might think divisive leaders like Donald Trump and Kim Jong-un would be encountering protests as they move about. Not in Singapore - a heavily controlled state. You can only protest with a permit (which is hard to get) and only in one particular park.

The areas around the hotels where the leaders were staying and meeting have also been placed under special security measures for the duration, meaning you can be searched by police on demand. You can't take in anything resembling a protest banner or items like loudhailers.

However, a small approved peace rally did take place over the weekend. Local journalist and activist Kirsten Han shared an image.

"South Korea will try to keep its improved ties with the North going," says BBC correspondent Robin Brant in Seoul. "For South Korea it's about easing military tensions, setting up a liaisons office, working on family reunions.

"So President Moon Jae-in has a separate track to keep the two Koreas closer regardless of what happens today in Singapore."

Image source, Getty Images

Yes, it’s not the first big deal Korea summit. In April, South Korea's President Moon Jae-in met with Kim Jong-un at the North-South border. At the end, they issued a declaration promising to:

• Keep talking, and build a "future of co-prosperity and unification"
• "Cease all hostile acts" against each other
• End the Korean War for good and realise "through complete denuclearisation, a nuclear-free Korean peninsula"

Don't forget, the Korean War ended with a truce - there was never a formal peace treaty.

"They had three handshakes, and all three times Trump reached out first," Manoj Vasudevan, a body language expert, told the BBC. He said that could be a way of Mr Trump asserting his dominance.

However, he added that Mr Trump "didn't receive the kind of handshake he was expecting".

"In American culture, [you tend to have] longer handshakes. He didn't receive that."

Image source, Getty Images

"Both sides have already declared this summit a success," BBC correspondent Rupert Wingfield Hayes explains, as Mr Trump and Mr Kim enjoy their working lunch behind closed doors.

"The one thing that's missing from all this is anything of substance. What are both sides putting on the table, what are the specifics? What does the US get from North Korea, and what it Washington offering in return?"

There were some quite surreal sights last night. Kim Jong-un took an unannounced (but clearly well-prepared and well-guarded) tour of Singapore's Marina Bay.

He walked around the tourist site for a few hours, guided by Singaporean officials and government ministers, and dropped in to the futuristic Gardens by the Bay tourist attraction.

We've got a gallery of some of the best pictures from that strange evening here.

Image source, Reuters

"It was only three months and three days ago that Donald Trump accepted Chairman Kim's invitation to talks. Only nine months since the US president called him Little Rocket Man, and Kim Jong-un called Donald Trump a 'mentally deranged dotard'.

"Now they are building a relationship and talking peace. For sure there will be massive obstacles to overcome, and many serious policy issues that could up-end hopes. But who'd have thought we'd be here?"

Laura Bicker, the BBC's Koreas correspondent, is in Singapore and writes:

"This is surely one of the most anticipated and remarkable handshakes in history. Even before the two men appeared there was something unthinkable about seeing the six North Korean flags flying alongside six Stars and Stripes on the white portico where they would meet.

"Then the two men emerged, earnest demeanours and cautious smiles. The president in trademark red tie, Mr Kim in equally trademark black Mao suit. The handshake, lasted 12 seconds, and then with their respective translators they walked to a nearby anteroom and exchanged pleasantries."

North Korea has long portrayed the US as public enemy number one, but conversations with ordinary North Korean citizens revealed that the government's attitude may be softening.

"Things are changing a little recently," Sun Hui, a North Korean citizen, told the BBC's Victoria Derbyshire programme ahead of the summit.

"Recently they say we should be living in peace with America, for everyone to have a better life."

Read more: Ordinary North Koreans speak out

One of the dishes on the summit lunch menu is unfamiliar even to South Koreans, tweets Minji Lee of BBC News Korean.

In case you missed the seismic moment, here it is. Kim Jong-un met Donald Trump at a hotel in Singapore. They went on to have a private chat of just over half an hour, then brief talks with their aides.

They're now at lunch.

If you're just waking up in Europe or on a lunch break in Asia, welcome to our coverage of the historic day here in Singapore. US President Donald Trump has met, shaken hands with and had talks with North Korean leader Kim Jong-un.

Why do we keep saying it's historic? Because no US president in office has ever met a North Korea leader. North Korea was shunned for decades, because it has an atrocious human rights record and has, to global consternation, developed nuclear weapons.

Only months ago, Mr Trump threw established US policy out the window and agreed to meet Mr Kim. Their handshake has, for better or worse, changed the course of history.

An extraordinary quote has emerged from the press pool, attributed to Kim Jong-un.

"The entire world is watching and many will see the unlikely meeting as something out of science fiction."

Su-Min Hwang, editor of BBC News Korean is at the press centre in Singapore. Here's her quick take on what today's sights mean for people in South Korea.