Globally hyperbolic spacetimes with a timelike boundary model spacetimes with naked singularities distributed on a timelike hypersurface, potentially admitting conditions on both the boundary and a Cauchy hypersurface. Typically, the boundary can serve as: (1) a cut-off for the system under consideration, or (2) a conformal completion of the spacetime, as in the case of asymptotically Anti-de Sitter spacetimes.

First, we will review some of their general properties, including those related to causality, the existence of global orthogonal splittings, and the relationship between the properties of the boundary and the interior of the spacetime. Then, we will focus on the null-convexity of the boundary. This is a natural assumption satisfied by asymptotically AdS spacetimes, which implies that the interior of the spacetime retains most of the geometric properties of the boundaryless case, such as being causally simple with a Hausdorff space of lightlike geodesics.

Based in joint work with L. Aké and JL. Flores (arxiv: 1808.04412) and with J. Herrera (arxiv: 2506.09032).