  We introduce and study the notions of hyperbolically embedded and very
rotating families of subgroups. The former notion can be thought of as a
generalization of the peripheral structure of a relatively hyperbolic group,
while the later one provides a natural framework for developing a geometric
version of small cancellation theory. Examples of such families naturally occur
in groups acting on hyperbolic spaces including hyperbolic and relatively
hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group.
Other examples can be found among groups acting geometrically on $CAT(0)$
spaces, fundamental groups of graphs of groups, etc. We obtain a number of
general results about rotating families and hyperbolically embedded subgroups;
although our technique applies to a wide class of groups, it is capable of
producing new results even for well-studied particular classes. For instance,
we solve two open problems about mapping class groups, and obtain some results
which are new even for relatively hyperbolic groups.
