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Illustrating the fascinating interplay between physics and mathematics, this book provides a solid grounding in the theory of groups, and particularly of group representations. The book will give the reader a firm grasp of the basics to enable further study as necessary. The approach taken allows the physics student to recognise the power and elegance of the abstract, axiomatic method. However, the formal mathematical definitions and proofs are always illustrated by concrete examples, and chapters developing the formalism are closely followed by chapters dealing with the physical applications. Problems are given after each chapter, with worked solutions at the end of the book. Intended for third year physics undergraduates specializing in theoretical physics but also of interest to mathematics undergraduates and first year postgraduates.