Let X be a compact Riemann surface. Fix a holomorphic line bundle L over X. A Hitchin pair (E,\phi) on X consists of a holomorphic vector bundle E and an endomorphism \phi of E twisted by L. We prove that (under certain conditions on the genus of X, the degree of L and the rank and degree of E) the moduli space of Hitchin pairs is irreducible, and that its isomorphism class uniquely determines the isomorphism class of the Riemann surface X. The talk is based on joint work with I. Biswas and M. Logares.