The theory of representations of loop groups provides a framework where one can consider Riemann surfaces with arbitrary numbers of handles and nodes on the same footing. Using infinite grassmanians we present a general formulation of some conformal field theories on arbitrary surfaces in terms of an operator formalism. As a by-product, one can obtain some general results for the chiral bosonization of fermions using the vertex operator representation of finite dimensional groups. We believe that this set-up provides the natural arena where the recent proposal of Friedan and Shenker of formulating string theory in the universal moduli space can be discussed.
