In the paper a model for optimal scheduling of inspections is considered, which can be used for the description of numerous situations in industry, military and medicine. First the existence of an optimal inspection strategy is shown under some weak presumptiuns.Then methods are developed allowing their computation. Finally minimax-inspection strategies are given for the following cases: 1) The distribution of the lifetime of the.element is entirely unknown, 2) a percentile of this distribution is known and 3) the mean value of the lifetime is known.