In this talk I will review a synthetic approach to Lorentzian geometry akin to Alexandrov- or CAT(k)-spaces in metric geometry. The goal is to have a notion of curvature (bounds) even if the underlying space is non-smooth or not a manifold. The talk will give some background for the talks of Robert McCann and Mathias Braun three weeks ago. Moreover, applications to General Relativity are given, in particular singularity theorems and (in)extendibility questions.