A well-known theorem of Hurwitz states that if τ[ratio ]S→S is a conformal self-mapping of a compact Riemann surface of genus g[ges ]2, then it has at most 2g+2 fixed points and that equality occurs if and only if τ is a hyperelliptic involution.

In this paper we consider this problem for a K-quasiconformal self-mapping f[ratio ]S→S. The result we obtain is that the number of fixed points (suitably counted) is bounded by 2+g(K1/2+K−1/2), and that this bound is sharp. We see that when K=1, that is, when f is conformal, our result agrees with the classical one.