2015 American Mathematical Society. We introduce a notion of hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings, natural pseudogroups acting on leaves of stable (or unstable) foliation of an Anosov diffeomorphism, etc. We describe a duality theory for hyperbolic groupoids. We show that for every hyperbolic groupoid G there is a naturally defined dual groupoid GT acting on the Gromov boundary of a Cayley graph of G. The groupoid GT is also hyperbolic and such that (GT)T is equivalent to G. Several classes of examples of hyperbolic groupoids and their applications are discussed.
