The purpose of this paper is to reprove some classical results on compact Riemann surfaces using elliptic operators, and to show how these results fit naturally into the general context of spin-manifolds. This programme was stimulated by discussion with D. Mumford and T am greatly indebted to him for information about the classical theory. Moreover, in [8], Mumford gives new algebraic proofs of the theorems in question, rather in the spirit of this paper.