The purpose of the paper is to undertake a detailed investigation of the role of Clifford algebras and spinors in the K&theory of real vector bundles. On the one hand the use of Clifford algebras throws considerable light on the periodicity theorem for the stable orthogonal group. On the other hand the use of spinors seems essential in some of the finer points of the KO-theory which centre round the Thorn isomorphism. As far as possible we have endeavoured to make this paper self-contained, assuming only a knowledge of the basic facts of Kand KO-theory, such as can be found in [3]. In particular we develop the theory of Clifford algebras from scratch. The paper is divided into three parts.

Witten [12] has interpreted the Donaldson invariants of four-manifolds by means of a topological Lagrangian. We show that this Lagrangian should be understood in terms of an infinite-dimensional analogue of the Gauss-Bonnet formula. Starting with a formula of Mathai and Quillen for the Thom class, we obtain a formula for the Euler class of a vector bundle, which formally yields the explicit form of Witten's Lagrangian. We use the same method to treat Lagrangians proposed for the Casson invariant.