This book's parent Topological Geometry (Porteous (1969)), originally written in the 1960's to make propaganda for a basis-free approach to the differential calculus of functions of several variables, contained, almost by accident, a central section on Clifford algebras, a generalisation of quaternions that was at that time little known. This section was strengthened in the second edition (Porteous (1981)) by an additional chapter on the triality outer automorphism of the group Spin(8), a feature which illuminates the structure of several of the other Spin groups and which is related to a property of six-dimensional projective quadrics first noticed almost a hundred years ago by Study in work on the rigid motions of three-dimensional space.

In recent years Clifford algebras have become a more popular tool in theoretical physics and it seems therefore appropriate to rework the original book, summarising the linear algebra and calculus required but expanding the Clifford algebra material. This seems the more worth while since it is clear that the central result of the old book, the classification of the conjugation anti-involution of the Clifford algebras Rp,q and their complexifications, was dealt with too briefly to be readily understood, and some of the more recent treatments of it elsewhere have been less than complete.

As in the previous version, the opportunity has been taken to give an exhaustive treatment of all the generalisations of the orthogonal and unitary groups known as the classical groups, since the full set plays a part in the Clifford algebra story.