Athletes from both sides, technically still at war, marched together at the opening ceremony and have fielded a unified women's ice hockey team, the first time an inter-Korean team has competed at any Olympic Games.
The personal invitation from Kim was delivered by his younger sister, Kim Yo Jong, during talks and a lunch Moon hosted at the presidential Blue House in Seoul.
Correspondents say the invitation puts Mr Moon in a hard position as he campaigned on a promise to engage with the North, but his moves in that direction go contrary to the wishes of his ally the US.
Tillerson was reacting to Vice President Mike Pence's remarks to the Washington Post on Sunday that the Trump administration is now willing to sit down and talk with North Korea. Moon shared details with Pence of his meeting with North Korean leaders, but did not talk about the invitation to talks in Pyongyang.
South Korean President Moon Jae-in, right, shook hands Saturday with Kim.
Always flanked by thick groups of bodyguards, Kim Yo Jong commanded attention wherever she went, walking among throngs of journalists with a quiet poise and occasionally shooting an enigmatic smile at cameras.
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A visit by Moon to the North would enable the first summit between leaders from the two Koreas since 2007.
Many experts say Kim's overture is meant to use improved ties with Seoul as a way to weaken USA -led worldwide sanctions toughened after its series of big weapons tests previous year that include its sixth and largest nuclear test explosion and three intercontinental ballistic missile launches.
Kim's presence at the Olympics has presented people in South Korea and beyond with something new to know about North Korea.
The offer was confirmed by South Korean presidential spokesman Kim Eui-kyeom at a press briefing.
North Korea is blamed for raising the ante by frequently resorting to nuclear and missile tests but of late it demonstrated its willingness to ease tension by deciding to participate in what is being hailed as "peace" Olympics in Pyeongchang in the South. However, the meeting could sow discord between Mr Moon and US President Donald Trump, who until recently, has exchanged personal insults and apocalyptic threats with Mr Kim. Her visit comes one year after the U.S. Treasury Department placed sanctions on Kim for her role in propaganda and "agitation" as part of her family's harsh rule over North Korea, and her government's efforts to develop nuclear weapons.
He will have one of two options: Start a war, which will guarantee his death, or stop his nuclear buildup and actually begin to use assets to treat the North Korean people decently, which will guarantee him life.