  We review some recent results on geometric equations on Lorentzian manifolds
such as the wave and Dirac equations. This includes well-posedness and
stability for various initial value problems, as well as results on the
structure of these equations on black-hole spacetimes (in particular, on the
Kerr solution), the index theorem for hyperbolic Dirac operators and properties
of the class of Green-hyperbolic operators.
