  We consider N=8 gauged supergravity in D=4 and D=5. We show one can weaken
the boundary conditions on the metric and on all scalars with $m^2 <-{(D-1)^2
\over 4}+1$, while preserving the asymptotic anti-de Sitter (AdS) symmetries.
Each scalar admits a one-parameter family of AdS-invariant boundary conditions
for which the metric falls off slower than usual. The generators of the
asymptotic symmetries are finite, but generically acquire a contribution from
the scalars. For a large class of boundary conditions we numerically find a
one-parameter family of black holes with scalar hair. These solutions exist
above a certain critical mass and are disconnected from the Schwarschild-AdS
black hole, which is a solution for all boundary conditions. We show the
Schwarschild-AdS black hole has larger entropy than a hairy black hole of the
same mass. The hairy black holes lift to inhomogeneous black brane solutions in
ten or eleven dimensions. We briefly discuss how generalized AdS-invariant
boundary conditions can be incorporated in the AdS/CFT correspondence.
