But on the eve of the summit, the White House unexpectedly announced Trump would depart Singapore by Tuesday evening, meaning his time with Kim would be fairly brief.
President Trump has said the best-case scenario is not going to resolve all these issues. "We will all know soon whether or not a real deal, unlike those of the past, can happen!"
After Trump's stern speech about North Korea's nuclear weapon program last November, Kim's government responded by calling the USA president an "old lunatic" - to which Trump responded by passively calling the North Korean leader "short and fat".
He would not say whether that included the possibility of withdrawing USA troops from the Korean Peninsula, but said the context of the discussions was "radically different than ever before". On the day before the meeting, weeks of preparation appeared to pick up in pace, with U.S. and North Korean officials meeting throughout Monday at a Singapore hotel.
But there was no mentioning the previous USA aim of "complete, verifiable and irreversible denuclearization".
Throughout the day, Trump and Kim's body language was openly friendly, a striking warmth given Kim's iron grip on power and dismal record on human rights.
In the same statement, the White House said the talks between U.S. and North Korean officials were moving "more quickly than expected", though didn't elaborate on what precisely that meant for Trump's stated goal of securing North Korea's complete and verifiable denuclearization.
Trump and his advisers have said the United States would accept nothing less than full denuclearization before sanctions against the North are lifted, and before North Korea can be fully welcomed into the fold of the broader diplomatic community.
Kim is due to leave on Tuesday afternoon, a source involved in the planning of his visit said on Sunday.
On Monday morning, the KCNA and the Rodong Sinmun, the official newspaper of the North's ruling party, gave extensive reports on Kim's departure for Singapore and his meetings with Singaporean officials, including its Prime Minister Lee Hsien Loong.
The new year began with another televised address from Kim, this time involving threats of a "nuclear button" on the North Korean leader's desk.
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The selfie posted by the Singaporean foreign minister shows a grinning Kim, donning his tortoise-shell glasses.
Giving voice to the anticipation felt around the world as the meeting opened, South Korean President Moon Jae-in said Tuesday he "hardly slept" before the summit.
Just a few months ago, Kim was an worldwide pariah accused of ordering the killing of his uncle, a half-brother and scores of officials suspected of disloyalty.
So the summit was almost derailed by talk from Trump's hawkish National Security Advisor John Bolton of a "Libya model" of disarmament. Immediately after the encounter, Dennis Rodman - one of the only Americans to have met Kim - was openly weeping while being interviewed by CNN's Chris Cuomo.
And in just a little while, this historic first will take place in Singapore.
It was a striking about-face from less than a year ago, when Trump was threatening "fire and fury" against Kim, who in turn scorned the American president as a "mentally deranged United States dotard".
His excursion out of the hotel caused a big stir with guests who were told to get behind a barricade while dozens of North Korean security men and Singapore police officers and government officials blocked off the lobby.
Some experts said Kim's city tour suggests that North Korea and the USA had narrowed their differences ahead of the summit. It heralded the summit as part of a "changed era". She also led a large North Korean art troupe sent to South Korea during the February Winter Olympics in Pyeongchang.
"Why would Kim... believe any commitments President Trump makes when he arbitrarily tears up an agreement with which the other party is complying?" said Antony Blinken, who was deputy secretary of state under Barack Obama. The two will later be joined by officials and have lunch together.
It is an outcome that seemed unimaginable less than a year ago - when Mr Trump and Mr Kim exchanged streams of fiery insults - and North Korea conducted several ballistic missile tests in defiance of the worldwide community.