  We study conformal blocks (the space of correlation functions) over compact
Riemann surfaces associated to vertex operator algebras which are the sum of
highest weight modules for the underlying Virasoro algebra. Under the fairly
general condition, for instance, $C_2$-finiteness, we prove that conformal
blocks are of finite dimensional. This, in particular, shows the finiteness of
conformal blocks for many well-known conformal field theories including WZNW
model and the minimal model.
