The armed forces of South Korea and Japan and US troops in the region were put on alert, though no unusual troop movements were monitored in North Korea. South Korea's President, Kim Young Sam, called an emergency cabinet meeting soon after the announcement of the death. He said people should remain vigilant, but there was no immediate cause for concern.
In the longer term, attention is focused on Kim Jong Il, 52, the son of Kim Il Sung, who for 20 years has been groomed to take over from his father. Dubbed the 'Dear Leader', Kim Jong Il is a mysterious and little-known figure, even by North Korean standards. Associated by some with vicious terrorist attacks against South Korean targets, including the blowing up of half the South Korean cabinet in Rangoon in 1983 and the bombing of an airliner over the Indian Ocean in 1987, Kim is thought to enjoy less than the full support of the military. It is unclear whether he will be able to maintain the same hold on power as his father did for 46 years.
A short man with a shrill voice, Kim Jong Il is given to wearing platform shoes, smoking American cigarettes and watching American films on video.
He has a reputation as a spoilt playboy. He studied in the former East Germany and has made at least one trip to China, but otherwise has little overseas experience, and rarely meets foreign dignitaries.
There seems little doubt that he will take over for the time being. Since the Great Leader's death was announced, the state radio has been mixing revolutionary music and readings from Kim Il Sung with extravagant praise for his son. 'Today, at the forefront of our revolution stands Comrade Kim Jong Il, great successor of the Juche (self-reliance) revolution, brilliant leader of the party and the people, and top commander of revolutionary armed forces,' said Pyongyang radio.
But North Korea watchers argue that if Kim Jong Il's position were secure, there would be no need for the outlandish propaganda which portrays him as a great scientist, philosopher, musician, industrial planner and protector of war orphans. The Chinese have long been opposed to his succession, though they have expressed their disapproval mainly by saying nothing about him.
The North Korean military has also been rumoured to be opposed to the Dear Leader. He has the official rank of marshal, is technically head of the million-strong armed forces, and is regularly cited as a military genius - though he has never actually served a day in the army.
Above all, it is unclear whether it is possible for someone to step into Kim Il Sung's shoes. Everyone in North Korea is ranked according to their level of loyalty to the Great Leader himself. There are three main classes, distinguished by colour-coded lapel badges: the 'core' class of most trusted revolutionaries; the 'wavering' class whose loyalty may be suspect; and the 'hostile' class which has been found to oppose the regime and is subjected to hard labour in work camps.
Now that Kim Il Sung is gone, it will be difficult to replicate the same structure by shifting loyalties onto anyone else, even his son.
But while South Korea worries about the prospect of a power vacuum that could lead to the collapse of the Communist North, with attendant huge reunification costs, the rest of the world is primarily concerned with North Korea's secretive nuclear programme.
Kim Jong Il has taken little part in the diplomatic manoeuvring among Pyongyang, Washington and the International Atomic Energy Agency in Vienna over the past 18 months, and his views on his country's suspected nuclear weapons programme are a mystery.
The talks between the US and North Korea about the nuclear issue, which began in Geneva on Friday, have been suspended. President Clinton has said that, after a period of mourning, he wants to see the talks reconvened.