Quasiconformal Homogeneity after Gehring and Palka

In a very influential paper Gehring and Palka introduced the notions of quasiconformally homogeneous and uniformly quasiconformally homogeneous subsets of R¯n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbb {R}}^n$$\end{document}. Their definition was later extended to hyperbolic manifolds. In this paper we survey the theory of quasiconformally homogeneous subsets of R¯n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbb {R}}^n$$\end{document} and uniformly quasiconformally homogeneous hyperbolic manifolds. We furthermore include a discussion of open problems in the theory.