The well-posedness of the Cauchy problem for symmetric hyperbolic systems on a Lorentzian manifold is a classical problem that has been thoroughly studied in many contexts. Particularly, if the underlying background is globally hyperbolic, a complete answer is known. Completely different is the situation if the Lorentzian manifold has a non-empty boundary. In this talk I will explain some recent results on the well-posedness of the Cauchy problem with (local) admissible boundary conditions. If time permits, I will also discuss the APS-boundary condition for the classical Dirac operator, as a first example of spatially non-local boundary condition. As this is a work in progress (with Nadine Große, Nicolas Ginoux and Nicolò Drago) I will present the motivations and expectations we have about our work.