In this talk I will describe a singular boundary value problem for Einstein metrics.  This problem arises in the Fefferman-Graham theory of conformal invariants, and in the AdS/CFT correspondence.   After giving a brief overview of some important results and examples, I will present a recent construction of boundary data which cannot admit a solution.  Finally, I will introduce a more general index-theoretic invariant which gives an obstruction to existence in the case of spin manifolds.  This is joint work with Q. Han and S. Stolz.