  We extend the modular orbits method of constructing a two-dimensional
orbifold conformal field theory to higher genus Riemann surfaces. We find that
partition functions on surfaces of arbitrary genus can be constructed by a
straightforward generalization of the rules that one would apply to the torus.
We demonstrate how one can use these higher genus objects to compute
correlation functions and OPE coefficients in the underlying theory. In the
case of orbifolds of free bosonic theories by subgroups of continuous
symmetries, we can give the explicit results of our procedure for symmetric and
asymmetric orbifolds by cyclic groups.
