The present volume is intended to serve a dual purpose. The first ten chapters are meant to be the basis for a course in Group Theory, and exercises have been included at the end of each of these chapters. The last ten chapters are meant to be useful as optional material in a course or as reference material. When used as a text, the book is intended for students who have had an introductory course in Modern Algebra comparable to a course taught from Birkhoff and MacLane's A Survey of Modern Algebra. I have tried to make this book as self-contained as possible, but where background material is needed references have been given, chiefly to Birkhoff and MacLane.