President Trump and Kim Jong Un have concluded their official schedule of meetings in Singapore, meetings Mr. Trump said he thinks went "better than anybody could have expected." Mr. Trump also said he and Kim will be "signing" some sort of document.

But it's unclear yet exactly what that means. A press conference is scheduled for 4 p.m. local time, or 4 p.m. EDT, in which more details will come forward.

As they began, Mr. Trump said he thinks he and Kim will have a "terrific relationship," and Kim, through a translator, said North Korea had to overcome a number of "obstacles" to get to this moment.

The stakes are high -- the U.S. seeks nothing less than the complete and verifiable denuclearization of the Korean peninsula, and Kim wants to see the sanctions on the North eased.

Mr. Trump and Kim participated in a one-on-one meeting, with translators only, followed by an expanded meeting including their top advisers, and a working lunch.

Here's Mr. Trump's full schedule for the day.

Follow Kim Jong Un-Trump summit live updates below:

Kim appears to check out Trump's vehicle

After Mr. Trump said they would be signing something, the president appeared to show Kim his vehicle.

It is unclear what is happening next.

Trump says he and Kim will be "signing" something

When Mr. Trump and Kim departed from their lunch, Mr. Trump said they will be "signing" something.

It is unclear what that might be, although Mr. Trump said the public will find out soon.

"Doing great. We had a really fantastic meeting," Mr. Trump told waiting reporters.

"A lot of progress," he continued. "Really very positive. I think better than anybody could have expected. Top of the line. Really good."

"We're going right now for a signing," Mr. Trump added.

Trump, Kim arrive at lunch — Trump has press take pictures so he looks "handsome"

Mr. Trump and Kim arrived at the lunch at just before noon local time, or midnight EDT, after their expanded bilateral meeting.

No U.S. media members were allowed in the brief pool spray, but the host TV captured Mr. Trump and Kim.

Mr. Trump suggested photographers get a picture so everyone looks "nice and handsome."

"Getting a good picture of everybody, so we look nice and handsome," Mr. Trump said.

But neither Mr. Trump nor Kim made any substantive comments about their conversation.

Here's the menu for the Trump-Kim lunch

The White House has provided the menu for the upcoming lunch between Mr. Trump and Kim. It mainly consists of Korean food -- with some Häagen-Dazs ice cream to top things off.

Starters

• Traditional prawn's cocktail served with avocado salad
• Green mango kerabu with honey lime dressing & fresh octopus
• "Oiseon"
• Korean stuffed cucumber

Main course

• Beef short rib confit, served with potato dauphinois and steam broccolini, red wine sauce on the side
• Combination of sweet & sour crispy pork and Yangzhou Fried Rice with homemade XO chili sauce
• "Daegu jorim"
• Soy Braised cod fish with radish, Asian vegetables

Desserts

• Dark chocolate tartlet ganache
• Häagen-Dazs vanilla ice cream with cherry coulis
• Tropezienne

Here are the people the White House says will be at the lunch:

• U.S. DELEGATION
• President Trump
• Secretary of State Mike Pompeo
• Chief of staff John Kelly
• National Security Adviser John Bolton
• Press Secretary Sarah Sanders
• Sung Kim, U.S. Ambassador to the Philippines
• Matthew Pottinger, deputy assistant to the president for Asian affairs

• NORTH KOREA DELEGATION
• Leader Kim Jong Un
• Kim Yong Chol, Vice-Chairman, Central Committee of the Workers' Party of Korea
• Ri Su Yong, Vice-Chairman, Central Committee of the Workers' Party of Korea
• Ri Yong Ho, Minister of Foreign Affairs
• No Kwang Chol, Minister of People's Armed Forces
• Choe Son Hui, Vice-Minister of Foreign Affairs
• Kim Yo Jong, First Deputy Director, Central Committee of Workers' Party of Korea
• Han Kwang Sang, Director, Central Committee of Workers' Party of Korea

Trump-Kim summit hits two-hour mark

Now that it's after 11 p.m. EDT, or 11 a.m. in Singapore, the Trump-Kim summit has reached the two-hour mark.

The expanded bilateral meeting is ongoing, and is still expected to last a while.

Who is Trump's translator?

Mr. Trump's translator, according to a U.S. State Department official, is Yun-hyang Lee, ‎Division Chief of Interpreting Services at the State Department.

Translators in a situation like this one, whether neither Mr. Trump nor Kim is fluent in the other's language and where the stakes are this high, are key.

Kim's translator: Many will see meeting as a "science fiction movie"

The translator for Kim to Mr. Trump, ahead of the expanded bilateral meeting, said many people will think of the summit "as a science fiction movie."

"Many people in the world will think of this as a (inaudible) form of fantasy...from a science fiction movie," the translator said.

Trump says that, "working together, we will get it taken care of"

Here is what was said at the top of an expanded bilateral meeting between Mr. Trump and Kim, and their top officials.

TRUMP: "Working together, we will get it taken care of."

KIM: "There will be challenges ahead but we will work with Trump. We overcame all kinds of skepticism and speculations about this summit and I believe that this is good for the peace."

TRUMP: "We will solve it...and I look forward to working on it with you. Thank you very much. Thank you very much everybody. Thank you."

Trump says meeting is "very, very good" so far

After their one-on-one meeting, Mr. Trump and Kim sat down for an expanded bilateral meeting.

In the room representing the U.S. are chief of staff John Kelly, Secretary of State Mike Pompeo and national security adviser John Bolton.

Ahead of that expanded bilateral meeting, Mr. Trump was asked how their summit is going so far.

"Very good," Mr. Trump said. "Very, very good. We have a great relationship. Thank you, thank you very much."

Trump and Kim complete first conversation

After about 35 minutes, at roughly 9:50 p.m., Mr. Trump and Kim emerged from the one-0n-one meeting.

They are now expected to proceed to an expanded bilateral meeting.

Trump and Kim speak, kicking off what Trump says will be a "terrific" relationship

Mr. Trump sat next to Kim and spoke ahead of their meeting.

"I feel really great," Mr. Trump told reporters, adding that it's his "honor" to meet and he and Kim will have a "terrific relationship."

"I feel really great, we're going to have a great discussion, and I think tremendous success," Mr. Trump said. "It will be tremendously successful. And its my honor, and we will have a terrific relationship I have no doubt."

Kim, through his translator, said there were a lot of obstacles to get to this moment, but they made it.

Only translators are expected to be in the meeting with Mr. Trump and Kim.

Trump, Kim shake hands for the first time

Mr. Trump and Kim shook hands for the first time just moments after 9 a.m. Mr. Trump did not smile as they shook hands, although Kim was spotted smiling after they greeted.

They shook hands for several seconds, before walking off towards their meeting together.

Trump, Kim have both arrived at summit site

By 9 p.m. EDT, both Mr. Trump and Kim had arrived at the site of the summit.

Trump tweets that top adviser Larry Kudlow suffered a heart attack

With moments to go before the Trump-Kim summit, the president tweeted that his top economic adviser, Larry Kudlow, has suffered a heart attack and is at Walter Reed National Military Medical Center in Washington, D.C.

Kim arrives on island where summit is taking place

According to the Associated Press, Kim has arrived on the island where the summit will take place in less than an hour's time.

At the summit site is a display of American and North Korean flags, where Kim and Mr. Trump are expected to shake hands.

Kim leaves his hotel for summit meeting

Kim has also departed his hotel en route to the summit, with dozens of vehicles making up his motorcade.

The summit is expected to start in less than an hour.

Trump leaves hotel for meeting with Kim

At just after 8 a.m. local time, Mr. Trump is on his way to the summit site.

As he departed, Mr. Trump tweeted about a Supreme Court ruling that came out hours before.

"Just won big Supreme Court decision on Voting! Great News!" Mr. Trump tweeted.

A live look at Trump and Kim's hotels as they prepare to depart for the meeting

Cameras are stationed outside the hotels where Mr. Trump and Kim are staying. They're expected to depart their respective locations at 8 p.m. ET for the drive to the Capella Hotel, where the meeting will take place:

Trump speaks with South Korean and Japanese leaders ahead of summit

The White House says the president spoke on the phone with South Korean President Moon Jae-in and Japanese Prime Minister Shinzo Abe on Monday ahead of the summit.

Here's how the White House describes the call with Moon:

President Donald J. Trump spoke yesterday with President Moon Jae-in of the Republic of Korea to discuss recent developments ahead of today's summit with North Korea. The two leaders vowed to continue their close coordination following President Trump's meeting with North Korean leader Kim Jong Un in Singapore.

And with Abe:

President Donald J. Trump spoke yesterday with Prime Minister Shinzo Abe of Japan. The two leaders discussed recent developments ahead of today's summit with North Korea and agreed to consult closely following the meeting.

What's at stake at the meeting?

Reporting from Singapore, CBS News chief White House correspondent Major Garrett has a look at what's at stake for both the U.S. and North Korea at the summit on tonight's "CBS Evening News":

Pompeo departs for summit meeting

The secretary of state is en route to the summit meeting, which is set to get underway in under two hours:

Trump tweets at "haters & losers" before meeting

A few hours before his meeting with Kim, the president acknowledged in a tweet that preliminary meetings "between staffs and representatives are going well and quickly," adding, "but in the end, that doesn't matter. We will all know soon whether or not a real deal, unlike those of the past, can happen!"

Then, he slapped at his critics, "the haters & losers" who say that "the fact that I am having a meeting [with Kim] is a major loss." He argued, "We have our hostages, testing, research and all missle [SIC] launches have stoped [SIC], and these pundits, who have called me wrong from the beginning, have nothing else they can say! We will be fine!"

What is success for the North Korea summit?

Pompeo told reporters on Monday that if diplomacy does not move in the right direction after Tuesday's talks, pressure will be ratcheted up on the North Korean regime. "We will see how far we get," Pompeo said of the negotiations.

Pompeo also emphasized that the U.S. has its eyes open, since it has been "fooled" before during prior attempts to reach peace in the region. He says that the "v" - verification -matters most.

What's up for discussion?

Both Mr. Trump and Kim are expected to largely focus their discussion on Tuesday around the denuclearization of the Korean Peninsula. Pompeo told reporters on Monday that the U.S. is prepared to offer "different, unique" security assurances to North Korea.

"We're prepared to take what will be security assurances that are different, unique than have been provided - than America has been willing to provide previously," Pompeo said. The secretary added that he's hopeful the summit will set the conditions for future productive talks and a potential framework for work towards denuclearization.

The president told Prime Minister Lee Hsien Loong of Singapore on Monday that he thinks "things could work out very nicely" during Tuesday's talks.