  We describe a self-consistent canonical quantization of Liouville theory in
terms of canonical free fields. In order to keep the non-linear Liouville
dynamics, we use the solution of the Liouville equation as a canonical
transformation. This also defines a Liouville vertex operator. We show, in
particular, that a canonical quantized conformal and local quantum Liouville
theory has a quantum group structure, and we discuss correlation functions for
non-critical strings.
