Trump and Kim met at a hotel on Sentosa, a popular tourist island a few hundred metres off the main island of Singapore.
The summit - the first between a sitting US president and a North Korean leader - will mark a turnaround of relations between Trump and Kim after a long-running exchange of threats and insults.
On the eve of the summit, the US offered "unique" security guarantees to North Korea in return for a "complete, verifiable and irreversible" denuclearisation.
The US insists it will accept nothing less than complete denuclearisation of the Korean peninsula.
The official North Korean news agency said on Sunday that Kim was ready to talk about "denuclearisation" and a "durable peace" at a summit held "for the first time in history under the great attention and expectation of the whole world."
Trump said on Saturday that Kim has a "one-time shot" to make history.
"I feel that Kim Jong-un wants to do something great for his people," he had said.
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