They are expected to discuss a roadmap for removing nuclear weapons from the Korean peninsula.
Little progress has been made on that and other issues since the first Trump-Kim summit in Singapore last year.
Wednesday's meetings were limited to brief questions from reporters, one-on-one talks and a "social dinner".
"Great meeting and dinner with Kim Jong-un," Mr Trump tweeted after the meal at the historic Metropole Hotel in Vietnam's capital, Hanoi.
Also at the table were US Secretary of State Mike Pompeo and Mr Kim's senior envoy, Kim Yong-chol.
What is on Thursday's agenda?
The two leaders are due to hold a series of meetings at the Metropole which will begin with another one-on-one session lasting 45 minutes, according to the White House.
A "joint agreement signing ceremony" will be held at the end of the meetings and Mr Trump is scheduled to give a news conference at 15:50 local time (08:50 GMT).
It is not clear what the joint agreement will include. Asked on Wednesday whether a formal end to the Korean War might be declared, Mr Trump said: "We'll see."
What happened on the first day?
Mr Trump and Mr Kim shook hands for reporters before holding talks and having dinner.
The US leader said he thought the summit would be "very successful" and denied he was "walking back" on denuclearisation.
"We had a very successful first summit," Mr Trump said, standing alongside Mr Kim. "Some people would like to see it go quicker - I'm satisfied, you're satisfied, we want to be happy with what we're doing."
Media captionDonald Trump and Kim Jong-un shook hands as they met in Hanoi
Mr Kim said he was confident there would be an "excellent outcome that everyone welcomes".
"I'll do my best to make it happen," he told reporters.
Separately, Mr Trump and Vietnamese President Nguyen Phu Trong signed trade deals worth around $20bn (£15bn). They included contracts for Vietnamese airline firms to buy US made planes and technology.
Why a second summit?
The two men met in Singapore last June in the first-ever summit between a North Korean leader and a sitting US president.
The Hanoi talks are expected to build on the groundwork of what was achieved then.
The Singapore meeting produced a vaguely worded agreement, with both leaders agreeing to "work towards denuclearisation" - though it was never made clear what this would entail.
Little diplomatic progress was made following the first summit. This time round, both leaders will be very conscious of the need to answer their critics with signs of concrete progress.
Washington had previously said that North Korea had to unilaterally give up its nuclear weapons before there could be any sanctions relief.
Media caption'I really miss our leader'
North Korean state media have praised Mr Kim for making the 4,000km (2,500-mile) trip, with state paper Rodong Sinmun dedicating four out of its six pages to it.
It said North Koreans had reacted to his visit with "boundless excitement and emotion", and urged people to work harder to "give him reports of victory when he returns".
The paper also added that his overseas trip had caused some of its citizens sleepless nights, with one woman telling a state broadcaster saying that she "really missed" Mr Kim.