The theory of group representations is a fundamental subject at the intersection of algebra, geometry and analysis, with innumerable applications in other domains of pure mathematics and in the physical sciences: chemistry, molecular biology and physics, in particular crystallography, classical and quantum mechanics and quantum field theory. Topics include: - brisk review of the basic definitions and fundamental results of group theory, illustrated with examples; - representation theory of finite groups (Schur’s Lemma and characters) and, using Haar measure, its generalization to compact groups; - Lie algebras and linear Lie groups; - detailed study of the group of rotations, the special unitary group in dimension 2 and their representations; - spherical harmonics; - representations of the special unitary group in dimension 3 (roots and weights) with quark theory as a consequence of the mathematical properties of the symmetry group. This book is an introduction to both the theory of group representations and its applications in quantum mechanics. Unlike many other texts, it deals with finite groups, compact groups, linear Lie groups and their Lie algebras, concisely presented in one volume. With only linear algebra and calculus as prerequisites, it is accessible to advanced undergraduates in mathematics and physics, and will still be of interest to beginning graduate students. Exercises for each chapter and a collection of problems with complete solutions make this an ideal text for the classroom or for independent study.