Since its start in 1980 the study of the consecutive-k system has resulted in the accumulation of hundreds of research papers. Its popularity is due to its close ties with many mathematical topics such that the system has become a prototype of how mathematical analysis can help in the study of system reliability. This is the first book to put together all the material on the subject. However, it is not just a collection of results. The authors have built a framework to fit the results into and then sort them and compare them, so that the reader has a good idea what is currently the best methodology. The authors also cover important extensions such as window systems, network systems, graph systems, and 2-dimensional systems. The consecutive-k system is known for its wide applicability and the authors have included a chapter on applications. Audience: All systems engineering researchers, not just those specializing in the consecutive-k system. The book could also be used for a graduate course to demonstrate how mathematics is actually applied to systems engineering.