Given a commuting d-tuple T Β―=(T 1 , ..., T d ) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator D T Β―. Significant attributes of the d-tuple are best expressed in terms of D T Β―, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T Β―derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, ...) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d-contractions of finite rank. It is shown that for the subcategory of all such T Β―that are (a) Fredholm and and (b) graded, the curvature invariant K(T Β―) is stable under compact perturbations. We do not know if this stability persists when T Β―is Fredholm but ungraded, although there is concrete evidence that it does.