  We study the CPT theorem for a two-dimensional conformal field theory on an
arbitrary Riemann surface. On the sphere the theorem follows from the
assumption that the correlation functions have standard hermiticity properties
and are invariant under the transformation $z \rightarrow 1/z$. The theorem can
then be extended to higher genus surfaces by sewing. We show that, as a
consequence of the CPT theorem on the world-sheet, the scattering $T$-matrix in
string theory is {\sl formally\/} hermitean at any loop order.
