This is a renewal-theoretic analysis of a class of single-item (s, S) inventory systems. Included, in a unified exposition, are both con tinuous and periodic review systems under fairly general random de mand processes. The monograph is complete in the sense that it starts from the derivation of the time dependent and stationary dis tributions of basic stochastic processes related to these systems and concludes with the construction and testing of simple, distribution free approximations for optimal control policies. However, it is rather incomplete as an account of single-item inventory systems in that it narrowly focuses on systems with full backlogging of unfilled demand and constant lead times, through what has come to be known as stationary analysis. The level is intermediate, and the style is informal. Some prior knowledge of probability theory and inventory control is assumed on the part of the reader. Given these, the monograph is self-contained. Extensive use is made ofrenewal-theoretic concepts and results; these are reviewed in Chapter 2.