A crucial element in the development of physical theories of nuclear and atomic structure, the theory of angular momentum can be applied with great effect to the formulation and solution of problems associated with the static magnetic and electric nuclear moments which are coupled to the electric and magnetic fields arising from surrounding charges. The theory relates to a variety of physical problems including those associated with low-temperature studies and microwave spectroscopy.This high-level treatment, based on a series of lectures the author gave at Oak Ridge National Laboratory, offers clear discussions of the general theory and its applications. Beginning with basic principles, the development is carried through the introduction of coupling coefficients for vector addition, the transformation properties of the angular momentum wave functions under rotations of the coordinate axes, irreducible tensors and Racah coefficients. Applications include static moments of systems composed schemes in nuclear reactions and more. In this volume, the author has attempted to achieve simplification by concerning himself only with the properties of rotations because of their intimate connection with the concept of angular momentum. In addition, the reasoning is inductive, and, as the theory initially develops, it makes a ""smooth-join"" with those aspects of quantum mechanics that are, comparatively speaking, common knowledge. A graduate-level knowledge of quantum mechanics is helpful for gaining maximum benefit from this highly regarded study.