Conformal groups play a key role in geometry and spin structures. This book provides a self-contained overview of this important area of mathematical physics, beginning with its origins in the works of Cartan and Chevalley and progressing to recent research in spinors and conformal geometry. Key topics:* The first chapter focuses on the basic material on Clifford algebras and spinor spaces, along with the necessary mathematical tools * The second chapter examines conformal spin geometry, starting with an elementary study of the classical conformal group in Euclidean space; it then treats covering groups of the conformal group and includes a section on the complex group * Conformal flat geometry and "conformal spinor groups" are introduced, and the text proceeds through a systematic development of manifolds endowed with conformal spin-structure and the theory of conformal connections* Chapter III covers conformal symplectic geometry and includes a discussion of conformal symplectic flat geometry and the covering groups of the real conformal symplectic group* The final chapter provides a detailed treatment of conformal pseudo-unitary spin structures, with emphasis on projective quadrics and pseudo-Hermitian spaces and their associated Clifford algebrasAll the major aspects of the field are introduced, followed by detailed descriptions and definitions; the books comprehensive index and bibliography give further directions for study and research. Rich in exercises that are accompanied by full proofs and many hints, this work will be ideal as a course text or as a self-study volume for senior undergraduates and graduate students. Its scope and clarity as a reference will also appeal to instructors and researchers in mathematics and physics.