  Extending the work of Park and Strominger, we prove a positive energy theorem
for the exactly solvable quantum-corrected 2D dilaton gravity theories. The
positive energy functional we construct is shown to be unique (within a
reasonably broad class of such functionals). For field configurations
asymptotic to the LDV we show that this energy functional (if defined on a
space-like surface) yields the usual (classical) definition of the ADM mass
{\it plus a certain ``quantum"-correction. If defined on a null surface the
energy functional yields the Bondi mass. The latter is evaluated careflly and
applied to the RST shock-wave scenario where it is shown to behave as
physically expected. Motivated by the existence of a positivity theorem we
construct manifestly supersymmetric (semiclassical) extensions of these
quantum-corrected dilaton-gravity theories.
