  This paper considers the existence of conformally compact Einstein metrics on
4-manifolds. A reasonably complete understanding is obtained for the existence
of such metrics with prescribed conformal infinity, when the conformal infinity
is of positive scalar curvature. We find in particular that general solvability
in this class depends on the topology of the filling manifold. The obstruction
to solvability for arbitrary boundary values is also identified. While most of
the paper concerns dimension 4, some general results on the structure of the
space of such metrics hold in all dimensions.
