The purpose of this book is to provide a treatment of these positive definite functions on abelian semigroups with involution. In doing so we also discuss related topics such as negative definite functions， completely monotone functions and Hoeffding-type inequalities. We view these subjects as important ingredients of harmonic analysis on semigroups. It has been our aim， simultaneously， to write a book which can serve as a textbook for an advanced graduate course， because we feel that the notion of positive definiteness is an important and basic notion which occurs in mathematics as often as the notion of a Hilbert space. The already mentioned Laplace and ourier transformations， as well as the generating functions for integer valued random variables， belong to the most important analytical tools in probability theory and its applications. Only recently it turned out that positive （resp. negative） definite functions allow a probabilistic characterization in terms of so-called Hoeffding-type inequalities.