In this talk, we will discuss the problem of uniquely continuing solutions to the Einstein and Klein-Gordon equations on asymptotically Anti-de Sitter spacetimes, from the conformal boundary. In particular, we will see how one can extend the result of Holzegel-Shao (arXiv:2207.14217), proving the unique continuation property for the Einstein vacuum equations, to the AdS-Einstein-Maxwell system. Such a result requires one to impose a geometric condition on the domain on which one imposes the boundary data, known as the Generalised Null Convexity Criterion. We will see how the failure of this condition implies non-uniqueness results for the Klein-Gordon equation. As illustrative examples, we will illustrate this phenomenon with the so-called planar AdS and pure AdS solutions.