The very mention of nuclear terrorism is enough to rouse strong reactions, and understandably so, because it combines the most terrifying weapons and the most threatening of people in a single phrase. The possibility that terrorists could obtain and use nuclear weapons deserves careful analysis, but discussion has all too often been contaminated with exaggeration, even hysteria. For example, it has been claimed that nuclear terrorism poses anexistential threat to the United States. This Adelphi Paper develops a more measured analysis of the risk of terrorists detonating a true fission device. The problem is attacked from two perspectives: the considerable, possibly insurmountable, technical challenges involved in obtaining a functional nuclear weapon, whetherhome-made or begged, borrowed or stolen from a state arsenal; and the question of the strategic, political and psychological motivations togo nuclear. The conclusions are that nuclear terrorism is a less significant threat than is commonly believed, and that, among terrorists, Muslim extremists are not the most likely to use nuclear weapons.