  We analyze actions for 2D supergravities induced by chiral conformal
supermatter. The latter may be thought as described at the classical level by
superspace actions invariant under super-reparametrization, super-Weyl and
super-Lorentz transformations. Upon quantization various anomalies appear which
characterize the non-trivial induced actions for the supergravitational sector.
We derive these induced actions using a chiral boson to represent the chiral
inducing matter. We show that they can be defined in a super-reparametrization
invariant way, but with super-Weyl and super-Lorentz anomalies. We consider the
case of $(1,0)$ and $(1,1)$ supergravities by working in their respective
superspace formulations and investigate their quantization in the conformal
gauge. The actions we consider arise naturally in off-critical heterotic and
spinning strings. In the conformal gauge, they correspond to chiral extensions
of the super-Liouville theory.
