  There exists an intriguing relation between genus zero correlation functions
in the H^+_3 WZNW model and in Liouville field theory. We provide a path
integral derivation of the correspondence and then use our new approach to
generalize the relation to surfaces of arbitrary genus g. In particular we
determine the correlation functions of N primary fields in the WZNW model
explicitly through Liouville correlators with N+2g-2 additional insertions of
certain degenerate fields. The paper concludes with a list of interesting
further extensions and a few comments on the relation to the geometric
Langlands program.
