Publisher Summary The chapter describes crystal field theory that concerns with an attempt to calculate as much as possible about the behavior of solids on the basis of a known or assumed symmetry of the problem and with the simplifying assumption that the interaction between the ion or atom of interest and its nearest neighbors is electrostatic in nature. Some recent group theoretical developments have made the application of group theoretical methods to solid state physics much more convenient. At the same time, physical problems have become much more complex. These two trends have combined to revive interest in group theoretical methods. Thus, it has seemed worthwhile to present the fundamental ideas of group theory and their application to crystal field theory from an advanced point of view, taking into account recent developments and problems. This chapter is on principles and theoretical methods, rather than results. Several topics of special relevance to crystal field theory are discussed—namely, time reversal and the Kramers theorem, and the Jahn-Teller theorem. With these preliminaries taken care of, the discussion of crystal field theory is straightforward.