Publisher Summary Every group A determines a group, Aut A, consisting of all automorphisms of A. This is commutative only in exceptional cases, so for the study of automorphism groups, noncommutative methods are inevitable. The chapter discusses the relationship between a group and its automorphism group. Because Aut A is nothing else than the group of units in E(A), it is evident that Aut A can provide less information only about A than collecting from E(A). On the other hand, concentration on Aut A means that one is able to make use of powerful group-theoretical methods; consequently, the approach must be different and one can expect to obtain new data in this way. The chapter presents an introductory discussion to get acquainted with the simple relations that exist among a group, its direct decompositions, and the automorphisms. A more comprehensive study of automorphism groups begins with the investigation of several normal subgroups in Aut A, closely related to A. More relevant results are presented in the chapter, where Aut A is thoroughly examined for torsion and torsion-free groups A, respectively.