Quasiconformal homeomorphisms are topological transformations with uniformly bounded distortion. This book concerns different notions of homogeneity; that is the degree of regularity of sets in extended Euclidean n-space which are assumed to have a family (or group) of quasiconformal homeomorphisms acting transitively. This builds on earlier important and interesting work, mainly in two dimensions, of other researchers. The novelty here is the generalisation to higher dimensions, where new tools and methods had to be developed. The techniques are quite complicated both analytically and topologically, because many classification theorems in two dimensions are not readily applicable in three or more dimensions. It is discovered in the book that there exists a subset of a domain in each dimension n which is quasiconformally homogeneous with respect to that domain but not even topologically homogeneous in Euclidean n-space.