  We consider bosonic supersymmetric backgrounds of ten-dimensional conformal
supergravity. Up to local conformal isometry, we classify the maximally
supersymmetric backgrounds, determine their conformal symmetry superalgebras
and show how they arise as near-horizon geometries of certain half-BPS
backgrounds or as a plane-wave limit thereof. We then show how to define
Yang-Mills theory with rigid supersymmetry on any supersymmetric conformal
supergravity background and, in particular, on the maximally supersymmetric
backgrounds. We conclude by commenting on a striking resemblance between the
supersymmetric backgrounds of ten-dimensional conformal supergravity and those
of eleven-dimensional Poincar\'e supergravity.
