The BBC spoke to analysts about the contenders and whether history is on their side.
A male member of the Kim family has been in charge of North Korea ever since its founding by Kim Il-sung in 1948 - and the mythology of this family runs deep throughout society.
Propaganda about its greatness begins for citizens before they can even read: pre-schoolers sing a song called: "I want to see our leader Kim Jong-un."
So how can you imagine a North Korea without this symbolic and political figure at the top? How would elites organise themselves, as well as society as a whole?
The easy answer is: we don't know. More interestingly, they don't know either. They have never had to do it.
As Kim Jong-un was being prepared for power, they even began using the term "Paektu Bloodline" to help legitimise his rule.
Paektu is the sacred and mythologised mountain where Kim Il-sung is said to have waged guerrilla war and where Kim Jong-il was reportedly born. Kim Jong-un still goes there when he wants to emphasise important policy decisions.
There has always been a Kim at the ideological heart of the country.
What would North Korea be like without such an heir? Kim Jong-un, 36, is believed to have children - but they are far too young. It is thought he has three children, the oldest being 10 and the youngest three. Kim Jong-un himself was considered young when he took power - he was 27.
It is likely that some sort of group leadership would emerge, perhaps as in Vietnam, that leans heavily on the founder's teachings and legitimacy to boost their own standing.
Observers can track who holds certain key positions and can follow news and open-source intelligence about important institutions, but can't really tell how factions are developing, nor who is holding power through personal rather than institutional bonds. Moreover, sometimes vice or deputy directors wield more real power than the titular heads of institutions. This makes all predictions extremely difficult.
The are three Kims who could potentially be involved in the political make-up of North Korea if Kim Jong-un were to disappear. They all face limitations in carrying on family rule.
The first is Kim Yo-jong, Kim Jong-un's younger sister. She is said to have been a favourite of her father who commented on her precocity, her interest in politics from a young age. Her manner is efficient, mild and one suspects rather observant. Much has been made of her closeness to her brother. At the Singapore Trump-Kim summit she was famously on hand to pass him a pen to sign the agreement with, and at the next summit in Hanoi, was pictured peeking out from behind corners as her brother posed for statesman-like photos.
Yet she was not above a temporary demotion after the Hanoi summit - purportedly because of its failure although this will never be confirmed. She doesn't sit on the top policy-making body, the State Affairs Commission, but is an alternate member of the Politburo and vice director of the Propaganda and Agitation Department (PAD) of the Workers' Party of Korea. These may seem like incomprehensible acronyms but the PAD is a powerful organisation that ensures ideological loyalty in the system.
She is a woman, however, and this makes it hard to imagine her occupying the top position in such a deeply patriarchal country. North Korea is an extremely male state, in which gender carries rigid expectations. Being supreme leader, and certainly running the military, does not fit in the range of womanly duties.
The second is Kim Jong-chul. He is Kim Jong-un's older brother, but has never appeared interested in politics or power. (He is known to be interested in Eric Clapton.) At most, he could be a symbolic link to the Kim family: perhaps made the head of a foundation and put forward to read the odd speech.
The final one is Kim Pyong-il, Kim Jong-il's half-brother. His mother - Kim Jong-il's stepmother - was angling to have him become Kim Il-sung's successor. She failed and was sidelined by Kim Jong-il as he rose in influence. Kim Pyong-il was sent to Europe in 1979, where he has held various ambassadorships, returning to North Korea only last year. This means it is very unlikely he has the network to be a central player in elite politics in Pyongyang.
There are other individuals who have been central in the Kim Jong-un era, but it is difficult to know who among them would form co-operative relationships and who would compete with one another.
One is Choe Ryong-hae. He has had his ups and downs under Kim Jong-un, but having weathered a few storms currently sits on the presidium of the politburo and is also first vice chairman of the State Affairs Commission. Last year he became the first new president in 20 years, replacing the aging Kim Yong-nam - so he is the person who represents the North at international engagements.
Choe has also held high positions in the military and the Organization and Guidance Department (OGD) of the Worker's Party of Korea, responsible for enforcing loyalty throughout the regime. This is an extremely powerful organisation: it enforces the adherence of all citizens to North Korea's ideology. He is probably the second most powerful man in North Korea.