This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and noncommutative. Aimed at students in mathematics, engineering, and the physical sciences. It examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research. With applications in chemistry error-correcting codes, data analysis, graph theory, number theory, and probability the book presents a concrete approach to abstract group theory through applied examples, pictures and computer experiments.