President Trump and North Korean leader Kim Jong Un signed a document on Tuesday stating that Pyongyang would work toward "complete denuclearization of the Korean Peninsula"— a historic concession, which was one of the requirements the U.S. sought at the summit in Singapore.

The historic agreement came after the two leaders held several meetings throughout the day. Trump was asked by a reporter if Kim agreed to denuclearize and he said, “We are starting that process very quickly.”

Trump did not refer to the document as a treaty or agreement.

Trump said at a press conference that he will be ending joint military exercises between the United States and South Korea.

He also said Kim agreed to destroy a 'major' missile testing site, but did not offer specific details.

The joint declaration states that the U.S. has committed to providing "security guarantees" to Pyongyang.

It's unclear exactly what Trump has promised Kim in terms of security. Secretary of State Mike Pompeo declined to say Monday whether guarantees might include withdrawing U.S. troops from the Korean Peninsula.

Col. Chad G. Carroll, U.S. Forces Korea spokesman, said there has been no updated guidance on execution or “cessation of training exercises.”

“We will continue with our current military posture until we receive updated guidance from the Department of Defense and/or Indo-Pacific Command,” Carroll said.

Trump appeared optimistic at a press conference after Kim had left the island.

Trump told reporters that the remains of U.S. prisoners of war and those missing in action during the Korean War will also be returned.

Trump said he asked Kim to commit to returning the remains "and we got it."

The president said he has received "countless calls" and letters from family members asking him to help them receive the remains of their loved ones.

A reporter asked Trump if he would be willing to invite Kim to the White House and he responded, “Absolutely I would.”

Trump was asked by reporters in Singapore during his final appearance with Kim on Tuesday what surprised him most during their meetings.

Trump said Kim has a "great personality" and is "very smart. Good combination."

Trump raised some eyebrows when he said he learned Kim is "a very talented man" and "loves his country very much."

Kim, who was sitting alongside Trump at an earlier conference, said through a translator, “We had a historic meeting and decided to leave the past behind and we are about to sign a historic document.”

Trump said future talks will include Secretary of State Mike Pompeo and national security adviser John Bolton. The joint agreement said North Korea will hold follow-up negotiations, led by Pompeo and "a relevant high-level DPRK official" at the "earliest possible date."

Reporters pressed Trump about human rights issues in the Hermit Kingdom, and the president acknowledged that the situation must improve.

Trump and Kim did not respond to a reporter who asked if they discussed Otto Warmbier.

Warmbier was an American student arrested in North Korean in January 2016 for stealing a propaganda poster and sentenced to 15 years in prison with hard labor. He was was sent home from North Korea in a coma and died soon after.

Trump later said that the meeting between the two leaders may not have happened if not for the death of Warmbier.

Trump said he thinks "we'll probably need another summit"-- or at least a second meeting. Trump said he is willing to meet with Kim "many times" in the process.

In the run-up to the talks, Trump had hopefully predicted the two men might strike a nuclear deal or forge a formal end to the Korean War in the course of a single meeting or over several days. But in a briefing with reporters Monday, Pompeo sought to keep expectations for the summit in check.

"We are hopeful this summit will help set the conditions for future productive talks," the secretary of state said.

Fox News' Samuel Chamberlain and The Associated Press contributed to this report