  In a very influential paper Gehring and Palka introduced the notions of
quasiconformally homogeneous and uniformly quasiconformally homogeneous subsets
of Euclidean space. Their motivation was to provide a characterization of
quasi-disks, i.e. domains which are quasiconformally homeomorphic to the unit
disk. As a generalization, Bonfert-Taylor, Canary, Martin and Taylor initiated
the study of uniformly quasiconformally homogeneous hyperbolic manifolds. In
this paper, we review the theory of quasiconformally homogeneous subsets of
Euclidean and uniformly quasiconformally homogeneous hyperbolic manifolds. We
finish with a discussion of open problems in the theory.
