  Scattering is defined on compact manifolds with boundary which are equipped
with an asymptotically hyperbolic metric, $g.$ A model form is established for
such metrics close to the boundary. It is shown that the scattering matrix at
energy $\zeta$ exists and is a pseudo-differential operator of order $2\zeta+1
- \dim X.$ The symbol of the scattering matrix is then used to show that except
for a countable set of energies the scattering matrix at one energy determines
the diffeomorphism class of the metric modulo terms vanishing to infinite order
at the boundary. An analogous result is proved for potential scattering. The
total symbol is computed when the manifold is hyperbolic or is of product type
modulo terms vanishing to infinite order at the boundary. The same methods are
then applied to studying inverse scattering on the Schwarzschild and De
Sitter-Schwarzschild models of black holes.
