Motivated by longstanding open question in the theory of the Einstein equations such as the strong cosmic censorship conjecture the talk will introduce a notion of Gromov-Hausdorff distance for a class of abstract Lorentzian spaces, called bounded Lorentz-metric spaces. I will discuss the definition and key examples as well as first stability results for natural geometric conditions under this notion of convergence. This is joint work with Ettore Minguzzi (Florence).