Discussion of further applications of the theory of Dirac operators and Clifford algebras now begins. The style of exposition will change somewhat, with fewer details being given than before, so that greater demands are placed upon the reader. A wider variety of topics can then be covered. In this chapter we shall discuss the representation theory for the group Spin(V, Q), concentrating almost entirely on the case of the compact group Spin(n) which arises when (V, Q) is an n-dimensional (real) positive- or negative-definite quadratic space (see also chapter 5). In the non-compact case, the characterization of the irreducible unitary representations is only just being discovered. But in the compact case these representations are all finite-dimensional and their ‘parameterization’ has been known for many years, thanks to the effort of Cartan, Weyl et al. This will be given in section 2 after some routine preliminaries have been disposed of in section 1. For detailed applications to analysis, however, various explicit realizations of these representations are needed. Representations of O(n), hence SO(n), and of Spin(n), on spaces of harmonic polynomials on ℝn are well-known, and their role in such singular integrals as Riesz transforms is well-understood. Realizations of more general representations of O(n) on harmonic polynomials on the space ℝr × n of real matrices are far less widely known; their use in singular integrals has hardly begun. In sections 4 and 5 we give a fairly detailed account of some aspects of the theory of polynomials of matrix argument, using it as a vehicle for presenting some aspects of polynomial invariant theory.