The Associated Press reports Trump was originally scheduled to depart early Wednesday morning, however he will now leave Tuesday night after a day of meetings with Kim and other leaders of North Korea. He said he would discuss it "at great length" at an upcoming news conference, and indicated it would be made available publicly.
"The world will see a major change", he said.
Kim said that "it was not an easy path here", but went on to say that "we've overcome everything and come to this place".
Before they signed the document, Trump and Kim had been involved in one-on-one talks, which came after months of planning.
Light on specifics, the document largely amounted to an agreement to continue discussions as it reiterated previous public statements and past commitments.
Nadal downs Del Potro to reach 11th French Open final
But the Spaniard was impressive on Friday, finishing with 35 winners and 19 unforced errors to Del Potro's analysis of 20 and 32. The Austrian reached the semifinal stage at this competition past year but is confident of going one better this time .
He reiterated that the goal of the summit remains the same: "the complete, and verifiable, and irreversible denuclearization of the Korean Peninsula".
The language of the agreement involves North Korea committing to complete denuclearization and long-lasting peace in exchange for unspecified "security guarantees" from the United States.
The joint declaration states that the US has committed to providing "security guarantees" to Pyongyang. "Anybody who takes over a situation like he did at 26 years of age and is able to run it, and run it tough - I don't say "he was nice" or I don't say anything about it", Trump said.
Mr Trump added: "I learned he's a very talented man".
He said that Donald Trump first caught his eye in 2007 when he won a "battle of the billionaires" in the World Wrestling Entertainment's (WWE) WrestleMania.
In the weeks since aides said she underwent a procedure to treat a benign kidney condition, Mrs Trump has directed her aides to release nearly no information about her medical condition or explain why she was at the hospital for five days.