We study properties of supergravity theories with non-compact gaugings, and
their higher-dimensional interpretations via consistent reductions on the
inhomogeneous non-compact hyperboloidal spaces {\cal H}^{p,q}. The gauged
supergravities are free of ghosts, despite the non-compactness of the gauge
groups. We give a general discussion of the existence of stationary points in
the scalar potentials of such supergravities. These are of interest since they
can be associated with de Sitter vacuum configurations. We give explicit
results for consistent reductions on {\cal H}^{p,q} in various examples,
derived from analytic continuation of previously-known consistent sphere
reductions. In addition we also consider black hole and cosmological solutions,
for specific examples of non-compact gaugings in D=4,5,7.Comment: Latex, 35 page
