The theorem connects the complex analysis of a compact Riemann surface with the surface’s topology, that is, with its genus. The theorem is named after Riemann and his student Gustav Roch (1839–66). There have since been important generalizations of the theorem by Atiyah and Singer and by Grothendieck.

As a holomorphic function on a compact Riemann surface is constant, it is the meromorphic functions that are of interest. If it is specified at certain points that a meromorphic function should have poles no worse than some given order, then the theorem gives information about the number of such meromorphic functions in terms of the specifications and the genus of the surface.