We discuss the initial boundary value problem in general relativity (with vanishing cosmological constant). We consider a non-standard set of boundary conditions, known as conformal boundary conditions, where the conformal class of the induced metric and the trace of the extrinsic curvature are fixed at the boundary. We compare these results with analogous results for the Dirichlet problem both in Lorentzian and Euclidean signature, where a notion of conformal black hole thermodynamics will be developed. Time permitting, we will discuss implications for holography and de Sitter space.