I will discuss some new uniqueness results for ill-posed Cauchy problems for semilinear wave equations. These results will depend on bounds on L^2 norms of the solution and its derivatives on constant time slices. First, I will discuss a result on Minkowski space that can be generalized to a class of curved spacetimes. The proof relies on a novel degenerate Carleman estimate. Then if there is time I will discuss a uniqueness result for the Klein-Gordon equation on the Schwarzschild spacetime utilizing energy estimates in a way pioneered by Morawetz.