"It’s a very great moment in the history of the world."

During a daylong summit in Singapore, President Donald Trump and North Korean leader Kim Jong Un signed an agreement to work toward complete denuclearization and a lasting "peace regime" on the Korean Peninsula.​

At a press conference following the meeting -- the first of its kind between a sitting US ​president and a North Korean leader -- Trump repeatedly touted its importance, at one point saying, "It’s a very great moment in the history of the world."

The president also said he was suspending ​military exercises on the Korean Peninsula and expected Kim to move "very quickly" to dismantle his nuclear arsenal. 

For his part, Kim said that the “world will see a major change” and that he and Trump “decided to leave the past behind."

However, the agreement they signed provided few details, and experts ​pointed out that the United States and North Korea have long understood the word ​"denuclearization" very differently.

Here are the major moments of the historic day.

Greeting: The two leaders met in person for the first time and shook hands before a line of American and North Korean flags. Trump ​predicted he and Kim would have a "terrific relationship" and that the summit would be "tremendously successful." 

Kim, through an interpreter, marveled that they had made it to the summit: "There were a lot of obstacles on the way here, but we overcame all of them."

Bi-lateral meeting: After Trump and Kim huddled privately for less than an hour, with only interpreters present, they moved to a larger meeting with key US and North Korea aides. 

Trump expressed confidence that he and Kim would get the "big dilemma" ahead of them "taken care of." 

Kim again referred to the past, saying, "The past worked as fetters on our limbs, and the old prejudices and practices worked as obstacles on our way forward. But we overcame all of them, and we are here today."

Working lunch: Trump and Kim ​sat for a working lunch. Trump jokingly asked the assembled camera operators if they were making him and Kim look "nice and handsome and thin." 

The menu may have given him cause for concern, given that it included: beef short rib confit, Korean stuffed cucumber, sweet and sour crispy pork with Yangzhou fried rice, and dark chocolate tartlet ganache for dessert.

Signing of the joint agreement: Seated at a long wooden table, with American and North Korean flags at their back, Trump and Kim signed the agreement. Trump called the ​document "very important" and "pretty comprehensive."

Trump's press conference: With Kim already aboard a flight back to North Korea, Trump addressed assembled press from around the world.

In an ebullient mood, Trump joked with reporters -- telling one they had the same hair -- and offered praise for Kim and their summit. 

The president went so far as to say he eventually wants to end the decades-long US military presence on the Korean Peninsula. Although Trump made clear that the withdrawal of the some 30,000 American troops in South Korea was not on the table at the moment, he said that he eventually hopes to bring them home. 

He also announced that he was suspending US military exercises on the Korean Peninsula. 

"At some point... I want to get our soldiers out. I want to bring our soldiers back home," he said. "But that's not part of the equation right now. At some point I hope it will be, but not right now."

Trump added: "We will be stopping the war games, which will save us a tremendous amount of money, unless and until we see the future negotiation is not going along like it should. But we'll be saving a tremendous amount of money. Plus, I think it's very provocative."

At the same time, Trump said economic sanctions against North Korea would stay in place.