This article examines the issue of when a State can be considered not to exercise jurisdiction under Article 1 of the European Convention on Human Rights in parts of its own territory. The starting point is that the notion of jurisdiction in the Convention is primarily territorial, and that a State is presumed to exercise jurisdiction throughout the entire territory. The presumption can be rebutted in exceptional circumstances. The author argues that this can happen only if the State is prevented from exercising authority and control in parts of the territory, and that the lack of authority and control must be caused by the influence of another State. Even if the State is considered to be in this position, the State retains a "reduced jurisdiction", and is under a positive obligation to take measures to protect the human rights of the population. The article examines the issue with particular reference to the Assanidze and Ilaşcu cases from the European Court of Human Rights, and illustrates the issue by reviewing briefly the current situation in four regions in the Caucasus.