The author introduces 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.

The author describes a duality theory for hyperbolic groupoids. He shows that for every hyperbolic groupoid $mathfrak{G}$ there is a naturally defined dual groupoid $mathfrak{G}^top$ acting on the Gromov boundary of a Cayley graph of $mathfrak{G}$. The groupoid $mathfrak{G}^top$ is also hyperbolic and such that $(mathfrak{G}^top)^top$ is equivalent to $mathfrak{G}$. Several classes of examples of hyperbolic groupoids and their applications are discussed.