In the absence of time-like boundary, the classical initial value problem in GR verifies a geometric uniqueness property. In particular, isometric Cauchy data leads to (maximal globally hyperbolic) developments which are isometric. While there exists several well-posed formulation of the initial boundary value problem in GR, no such geometric uniqueness is known. This important issue was put forward by Helmut Friedrich. It is relevant not only for the local initial boundary value problem, but also for more global aspects, due to the possible breakdown of gauge choices. I will review the mathematical analysis of the initial boundary value problem in GR, with an emphasis on various aspects relevant to the geometric uniqueness problem. If time (and progress) permits, I will present work in progress with Grigorios Fournodavlos concerning an approach to the initial boundary value problem based on the wave equation satisfied by the second fundamental form of a foliation with prescribed mean curvature.