US President Donald Trump has said Kim Jong-Un is a "great leader" as the pair met in Hanoi for the first time since their summit in Singapore last year.
The two had a one-on-one conversation at the Metropole Hotel at 6.30pm local time (11.30am Irish time), before having a "social dinner."
On Thursday, there will be a full summit of "back and forth meetings", according to the White House.
Both leaders arrived in the Vietnamese capital on Tuesday: Mr Kim in the morning, on an armoured train he took from Pyongyan; and Mr Trump in the evening, aboard Air Force One.
Mr Trump said last year that he and Mr Kim "fell in love."
Speaking after their meeting this morning, President Trump said it was an "honour" to be with Chairman Kim
"We had a very successful first summit," he said.
"Some people would like to see it go quicker but I am satisfied and you are satisfied; we want to be happy with what we are doing.
"I feel like the first one was a great success and I think this one, hopefully, will be equal or greater than the first
"We made a lot of progress and I think the biggest progress was our relationship, is really a good one."
He said he believes North Korea has tremendous economic potential and said Mr Kim was a "great leader."
"As I have said many times [...] I think that your country has tremendous economic potential," he said. "Unbelievable, unlimited."
"I think that you will have a tremendous future with your country - a great leader.
"I look forward to watching it happen and helping it to happen. And we will help it to happen."
Since their meeting in Singapore, they have exchanged letters via envoys on a semi-regular basis.
Despite that, little progress has been made towards the process of denuclearisation, which both leaders agreed to work towards in Singapore.
Mr Trump is convinced that his personal relationship with Kim, combined with prospect of economic prosperity for North Korea, can break the impasse.
He has tweeted: "Vietnam is thriving like few places on earth. North Korea would be the same, and very quickly, if it would denuclearise.
"The potential is AWESOME, a great opportunity, like none other in history, for my friend Kim Jong-Un. We will know fairly soon - Very interesting!"
The White House has pointed to the absence of missile launches or nuclear tests by North Korea as proof of progress, but there is pressure on the summit to produce specific concessions and perhaps a road-map with steps for denuclearisation.
From North Korea, these could include destroying its nuclear fissile material production facilities at Yongbyon and elsewhere, and returning more remains of American soldiers killed or captured during the Korean War.
In return, the US could offer to open liaison offices in North Korea and to foster cultural exchanges.
It could also declare an end to the Korean War. This would be a political settlement, and not replace the armistice that has held since 1953.
As a result, North and South Korea technically remain at war.
But it would be a first step towards a comprehensive peace settlement on the Korean Peninsula.
Mr Trump could also offer some limited sanctions relief.
Before his dinner with Kim, Mr Trump spent the day meeting the president and then the prime minister of Vietnam.
Mr Kim's schedule is not known. On Tuesday, he visited the North Korean embassy in Hanoi.
North Korean state media have been reporting Mr Kim's journey prominently.
Wednesday's edition of Rodong Sinmun, a state-run newspaper, said that the absence of Kim Jong-Un is leading to "sleeplessness" amongst North Korean workers and thus increased production.
Additional reporting Michael Staines