  Green-hyperbolic operators are linear differential operators acting on
sections of a vector bundle over a Lorentzian manifold which possess advanced
and retarded Green's operators. The most prominent examples are wave operators
and Dirac-type operators. This paper is devoted to a systematic study of this
class of differential operators. For instance, we show that this class is
closed under taking restrictions to suitable subregions of the manifold, under
composition, under taking "square roots", and under the direct sum
construction. Symmetric hyperbolic systems are studied in detail.
