This classic, concise text has served a generation of physicists as an exceptionally useful guide to the mysteries of angular momenta and Clebsch-Gordon Coefficients. Derived from notes originally prepared to assist graduate students in reading research papers on atomic, molecular, and nuclear structure, the text first reviews the basic elements of quantum theory. It then examines the development of the fundamental commutation relations for angular momentum components and vector operators, and the ways in which matrix elements and eigenvalues of the angular momentum operators are worked out from the commutation relations.These chapters constitute a review on an elementary level of material usually included in a one-year course in quantum theory. In the second half of the text, the matrix elements of scalar, vector, and tensor operators are computed and applied to derive several useful relations in the theory of magnetic moments, electric quadruple moments, and dipole transition probabilities. Eschewing all that is irrelevant, this book focuses on the essentials and prepares readers for further study in the field. A helpful list of references concludes the work.