  We study supersymmetric domain walls in N=1 supergravity theories, including
those with modular-invariant superpotentials arising in superstring
compactifications. Such domain walls are shown to saturate the Bogomol'nyi
bound of wall energy per unit area. We find \sl static \rm and \sl reflection
asymmetric \rm domain wall solutions of the self-duality equations for the
metric and the matter fields. Our result establishes a new class of domain
walls beyond those previously classified. As a corollary, we define a precise
notion of vacuum degeneracy in the supergravity theories. In addition, we found
examples of global supersymmetric domain walls that do not have an analog when
gravity is turned on. This result establishes that in the case of extended
topological defects gravity plays a crucial, nontrivial role.
