We address the possibility of false vacuum decay in $N=1$ supergravity theories, including those corresponding to superstring vacua. By establishing a Bogomol'nyi bound for the energy density stored in the domain wall of the $O(4)$ invariant bubble, we show that supersymmetric vacua remain absolutely stable against false vacuum decay into another supersymmetric vacuum, including those from a Minkowski to an anti-deSitter (AdS) one. As a consequence, there are no compact static spherical domain walls, while on the other hand there exist planar domain walls interpolating between non-degenerate supersymmetric vacua, e.g. between Minkowski (topology $\Re^{4}$) and AdS (topology $S^{1}(time) \times \Re^{3}(space)$) vacua. (Talk presented at the XXVI International Conference on High Energy Physics August 6-12, 1992, Dallas, Texas)

6 pages, UPR-529-T