It is well known that the problem of prescribing initial data on the boundary of a domain of dependence $\DD$ of solutions to a wave equation is not well posed in the complement of $\DD$. It is expected, however, that one still has uniqueness. In collaboration with Alexandru Ionescu we have been recently able to prove some uniqueness results both in the Minkowski space, as well as for the Schwarzschild and Kerr space-times in the domain of outer communication.