In Pyongyang, Pompeo and Kim met for almost four hours, first in a business session and then in a 90-minute luncheon that the North Korean leader hosted at a state guest house.
South Korea's presidential office later said that Kim and Pompeo had agreed to arrange a second summit between the North and the U.S. "as soon as possible".
Trump is pressing to meet with Kim for a second time after their June summit in Singapore produced a vague agreement on denuclearization with few, if any, specifics.
The North has committed to taking additional denuclearization steps, such as shutting down its long-range missile engine test site and launching pad in Dongchang-ri on the west coast in the presence of worldwide inspectors and closing its main Yongbyon nuclear complex should the US take "corresponding measures".
"It's a very nice day that promises a good future", Kim said, "for both countries".
North Korea's state newspaper, Rodong Sinmun, featured eight photos of the meeting on its front page, including shots of Kim and Pompeo smiling and shaking hands, as well as some with Kim's sister Kim Yo Jong.
"We are starting to see a first wave of actions we can take on all four pillars of the Singapore communique", said Biegun, who will serve as Pompeo's pointman in negotiations with the North as well as work with South Korea, Japan, China, Russia and other countries that have an interest in the talks.
Pompeo, who was making his fourth visit to Pyongyang, said on Twitter on Friday he would "continue our efforts to build out a pathway for the denuclearization of (North Korea)".
Trump and Kim met in Singapore in June, and that summit ended with Kim's pledge to denuclearize the Korean Peninsula.
During a closed meeting with Moon, Pompeo said he and Kim agreed to arrange the second U.S.
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Pompeo responded: "We had a great, great visit this morning". Pompeo also promised Abe he would bring up the issue of Japanese citizens abducted by North Korea in his meetings with North Korean officials. "Thanks for hosting me and my team".
"If you do see concrete action that assures the US, the rest of the world that definitely concrete steps are being taken to eliminate very important parts of their nuclear program, then that's a definite step forward and builds the trust", Kang told The Washington Post in an interview last week.
Secretary of State Mike Pompeo's trip to Asia has begun. Tension between the two governments also surfaced at last month's U.N. Security Council meeting when President Trump accused the Chinese government of trying to interfere in the U.S.'s upcoming midterm elections.
She also said she believed the two countries "wouldn't be here without Trump".
Pompeo's trip to Seoul, where he will meet with South Korean President Moon Jae-in, will likely involve reviewing negotiations with North Korea.
Pompeo also discussed additional denuclearization steps the reclusive regime can take, as well as Washington's monitoring of the verification process and "corresponding measures", according to Yoon.
After a previous visit to Pyongyang in July, Pompeo had said the two foes made progress on key issues - but within hours of his departure the North condemned "gangster-like" demands from the United States, raising questions over how much the two sides really saw eye to eye.
North and South Korea want a "phased" approach with Pyongyang rewarded as it it happens.
The United Nation's nuclear watchdog, the International Atomic Energy Agency (IAEA), say they are best placed to verify a dismantlement of North Korea's nuclear programme.
But several experts told ABC News that North Korea hadn't done enough yet to earn an end-of-war declaration, let alone a second Trump-Kim summit. Pompeo will continue traveling to Beijing and then return to Washington on Monday.