  The constraints of the superfield method in two-dimensional supergravity are
adapted to allow for nonvanishing bosonic torsion. As the analysis of the
Bianchi identities reveals, a new vector superfield is encountered besides the
well-known scalar one. The constraints are solved both with superfields using a
special decomposition of the supervielbein, and explicitly in terms of
component fields in a Wess-Zumino gauge. The graded Poisson Sigma Model (gPSM)
is the alternative method used to construct supersymmetric gravity theories. In
this context the graded Jacobi identity is solved algebraically for general
cases. Some of the Poisson algebras obtained are singular, or several
potentials contained in them are restricted. This is discussed for a selection
of representative algebras. It is found, that the gPSM is far more flexible and
it shows the inherent ambiguity of the supersymmetric extension more clearly
than the superfield method. Among the various models spherically reduced
Einstein gravity and gravity with torsion are treated. Also the Legendre
transformation to eliminate auxiliary fields, superdilaton theories and the
explicit solution of the gPSM equations of motion for a typical model are
presented. Furthermore, the PSM field equations are analyzed in detail, leading
to the so called "symplectic extension". Thereby, the Poisson tensor is
extended to become regular by adding new coordinates to the target space. For
gravity models this is achieved with one additional coordinate. Finally, the
relation of the gPSM to the superfield method is established by extending the
base manifold to become a supermanifold.
