  We construct rigid supersymmetric gauge theories on Riemannian
five-manifolds. We follow a holographic approach, realizing the manifold as the
conformal boundary of a six-dimensional bulk supergravity solution. This leads
to a systematic classification of five-dimensional supersymmetric backgrounds
with gravity duals. We show that the background metric is furnished with a
conformal Killing vector, which generates a transversely holomorphic foliation
with a transverse Hermitian structure. Moreover, we prove that any such metric
defines a supersymmetric background. Finally, we construct supersymmetric
Lagrangians for gauge theories coupled to arbitrary matter on such backgrounds.
