An introductory text book for graduates and advanced undergraduates on group representation theory.It emphasizes group theory's role as the mathematical framework for describing symmetry properties of classical and quantum mechanical systems.Familiarity with basic group concepts and techniques is invaluable in the education of a modern-day physicist.This book emphasizes general features and methods which demonstrate the power of the group-theoretical approach in exposing the systematics of physical systems with associated symmetry.Particular attention is given to pedagogy.In developing the theory, clarity in presenting the main ideas and consequences is given the same priority as comprehensiveness and strict rigor.To preserve the integrity of the mathematics, enough technical information is included in the appendices to make the book almost self-contained.A set of problems and solutions has been published in a separate booklet.