This paper critically reviews the bounds developed for the probability of union of events in structural reliability theory and other disciplines. The aim of this study is to show how the bounds developed in other fields can be directly used in structural reliability theory and to highlight the areas in which further research would be beneficial. It should be emphasised that the reliability problem is considered here only as one of probability theory: It is assumed that the probabilities P(Fi) and P(Fi∩Fj) are all properly defined and that methods are available for their evaluation. Some ideas are introduced to improve the linear bounds by the inclusion of higher order intersections. An improved lower bound has been developed using nonlinear combinations of P(Fi∩Fj) in place of P(Fi∩Fj∩Fk). Finally a computer algorithm based on Hunter's idea is presented for the evaluation of the upper bound.