Boundary value problems for the Dirac operator on a Riemannian manifolds are rather well understood. In particular, one has a general description of admissible boundary conditions. The Lorentzian case has been studied only recently and it turns out that there are similarities but also fundamental differences to the Riemannian case. I will describe both situations and contrast them. This is joint work with L. Bandara, W. Ballmann, S. Hannes and A. Strohmaier.