  We generalize a version of small cancellation theory to the class of
acylindrically hyperbolic groups. This class contains many groups which admit
some natural action on a hyperbolic space, including non-elementary hyperbolic
and relatively hyperbolic groups, mapping class groups, and groups of outer
automorphisms of free groups. Several applications of this small cancellation
theory are given, including to Frattini subgroups and Kazhdan constants, the
construction of various \exotic"' quotients, and to approximating
acylindrically hyperbolic groups in the topology of marked group presentations.
