  We study maximally supersymmetric solutions of all gauged or deformed
supergravity theories in $D \ge 3$ space-time dimensions. For vanishing
background fluxes the space-time background has to be either Minkowski or
anti-de Sitter. We derive a simple criterion for the existence of solutions
with non-trivial fluxes and determine all supergravities that satisfy it. We
show that their solutions coincide with those of the corresponding ungauged
theories and conclude that the known list of maximally supersymmetric solutions
is exhaustive.
