This book is a clear and self-contained introduction to the theory of groups. It is written with the aim of stimulating and encouraging undergraduates and first year postgraduates to find out more about the subject. All topics likely to be encountered in undergraduate courses are covered. Numerous worked examples and exercises are included. The exercises have nearly all been tried and tested on students, and complete solutions are given. Each chapter ends with a summary of the material covered and notes on the history and development of group theory. The themes of the book are various classification problems in (finite) group theory. Introductory chapters explain the concepts of group, subgroup and normal subgroup, and quotient group. The Homomorphism and Isomorphism Theorems are then discussed, and, after an introduction to G-sets, the Sylow Theorems are proved. Subsequent chapters deal with finite abelian groups, the Jordan-Holder Theorem, soluble groups, p-groups, and group extensions. Finally there is a discussion of the finite simple groups and their classification, which was completed in the 1980s after a hundred years of effort.