  Quantization of two-dimensional dilaton gravity coupled to conformal matter
is investigated. Working in conformal gauge about a fixed background metric,
the theory may be viewed as a sigma model whose target space is parameterized
by the dilaton $\phi$ and conformal factor $\rho$. A precise connection is
given between the constraint that the theory be independent of the background
metric and conformal invariance of the resulting sigma model. Although the
action is renormalizable, new coupling constants must be specified at each
order in perturbation theory in order to determine the quantum theory. These
constants may be viewed as initial data for the beta function equations. It is
argued that not all choices of this data correspond to physically sensible
theories of gravity, and physically motivated constraints on the data are
discussed. In particular a recently constructed subclass of initial data which
reduces the full quantum theory to a soluble Liouville-like theory has energies
unbounded from below and thus is unphysical. Possibilities for modifying this
construction so as to avoid this difficulty are briefly discussed.
