I summarize recent results on causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular di fferentiability assumptions. I also introduce the concept of closed Lorentz-Finsler space and explain why it provides the best definition of Lorentz-Finsler space under low regularity. Finally, I mention two applications: the proof of the Lorentz-Finsler version of Connes' distance formula and the characterization of the Lorentzian submanifolds of Minkowski spacetime.