In this talk I will consider the global rigidity of bar-joint frameworks in $\mathbb{R}^3$ whose vertices are constrained to lie on a fixed surface. When the surface is a plane or a sphere then generic global rigidity depends only on the graph and the properties of the graph guaranteeing uniqueness (or global rigidity) can be tested efficiently. I will describe these results and then talk about recent work to extend these characterisations to other surfaces.