Trump, in a post lunch walk with Kim around the grounds of the Capella Hotel, told reporters they were "heading for a signing".
Kim and his regime have made no secret that nuclear weapons are the key to North Korea's survival yet Mr Trump has made equally clear that he will not tolerate the development of a deadly arsenal that could strike the U.S. mainland.
When the US president meets with leaders of countries with dimmer views of free speech, the American role is typically to push for more access.
Who are the key players in the North Korea regime and how are they related to the Supreme Leader?
In a choice that seems to make light of a lot of really quite serious things at once, Twitter is promoting its new #TrumpKimSummit emoji for Tuesday's summit in Singapore.
Robert King, who served as USA envoy on North Korean human-rights issues under the Obama administration, said Trump has used human rights as an instrument to get Kim Jong Un to negotiate on nuclear weapons, but not as a policy priority in its own right.
IHOP flips over burgers, changes ‘p’ to ‘b’ in logo
In response to the burns, IHOb responded with "We don't want any beef with you, we just want to share our beef with the world". When asked if IHOP was considering food-fusion menu items - remember Burger King's Whopperito or Taco Bell's Nacho Fries?
But other reports said it was Kim who chose to leave early, even though Trump was willing to stick around. In that first meeting they were joined only by translators, a break from standard practice of having at least one aide present for high-stakes huddles.
One US official said the two sides were working to craft a joint US-North Korea "statement of understanding" that could be released after the talks.
"But despite of all that, we've made it here", said Mr Kim.
UPDATED | US President Donald Trump said he had forged a "good relationship" with North Korean leader Kim Jong-un at the start of a historic summit in Singapore, as the two men sought ways to end a nuclear stand-off on the Korean peninsula.
Trump has taken keen interest in the pageantry of the day, insisting the pictures beamed around the world reflect a commanding leader making a decisive, world-altering move.
Any nuclear deal will hinge on North Korea's willingness to allow unfettered outside inspections of the country's warheads and nuclear fuel, much of which is likely kept in a vast complex of underground facilities. That could generate misunderstandings and a breakdown that risks armed conflict over North Korea's nuclear weapons.
The spokesman said Trump told Moon he would send Pompeo to Seoul immediately after the summit to explain its outcome.
This was the first meeting between the two leaders - one a 71-year-old billionaire businessman democratically elected to head a superpower; the other a boyish heir handed the reins of a brutally oppressive dictatorship. And I believed in North Korea.
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