The monograph is devoted to the mathematical apparatus of the quantum theory of angular momentum, systematized and presented within the framework of a single system of definitions and designations. Along with the basic theoretical provisions, the monograph contains a large number of formulas and ratios necessary for calculations in the field of atomic and molecular physics, nuclear physics and elementary particle theory. The presented apparatus is also widely used in solving practical problems in the field of quantum chemistry, physical kinetics, plasma physics, quantum optics, microelectronics and spintronics.