This book is written for students of theoretical solidstate physics, and for experimental physicists who have an interest in the study of electronic states in crystals. The presentation emphasises those aspects of the theory which are of general validity and the mathematical methods employed arc applicable to any crystal-lographic type. The treatment does not assume an extensive knowledge of mathematics. To help the reader, for example, the equations required for the construction of character tables are derived in the text from a few basic theorems. The examples used to illustrate the spectra of electronic states in crystals are chosen for their practical importance rather than for convenience of form. The representation of electronic states in extended wave-vector space is discussed and methods are given for the reduction of this space to the fundamental Brillouin zone. This representation is applied to the electron theory of alloys. Spin-orbit fine structure in semi-conductors and the properties of the double group are considered in the final chapter.