Orthogonal, symplectic and unitary representations of finite groups lie at the crossroads of two more traditional subjects of mathematics--linear representations of finite groups, and the theory of quadratic, skew symmetric and Hermitian forms --and thus inherit some of the characteristics of both. This book is written as an introduction to the subject and not as an encyclopaedic reference text. The principal goal is an exposition of the known results on the equivalence theory, and related matters such as the Witt and Witt-Grothendieck groups, over the “classical” fields--algebraically closed, real closed, finite, local and global. A detailed exposition of the background material needed is given in the first chapter. It was A. Frohlich who first gave a systematic organisation of this subject, in a series of papers beginning in 1969. His paper Orthogonal and symplectic representations of groups represents the culmination of his published work on orthogonal and symplectic representations. The author has included most of the work from that paper, extending it to include unitary representations, and also providing new approaches, such as the use of the equivariant Brauer-Wall group in describing the principal invariants of orthogonal representations and their interplay with each other.