Metrically homogeneous spaces

This survey discusses the problem of describing properties of the class of metric spaces in which the Uryson construction of a universal homogeneous metric space (for this class) can be carried out axiomatically. One of the main properties of this kind is the possibility of gluing together two metrics given on closed subsets and coinciding on their intersection. The uniqueness problem for a (countable or complete) homogeneous space universal in a given class of metric spaces is discussed. The problem of extending a Clifford translation of a compact subset of an (ultrametric) Uryson space to a Clifford translation of the entire Uryson space is studied.