Kaleidoscopical configurations

Let G be a group, and let X be a G-space with the action G × X → X, (g, x) ?gx. A subset A of X is called a kaleidoscopical configuration if there is a coloring χ : X → κ (i.e., a mapping of X onto a cardinal κ) such that the restriction χ|gA is a bijection for each g ∈ G. We survey some recent results on kaleidoscopical configurations in metric spaces considered as G-spaces with respect to the groups of their isometries and in groups considered as left regular G-spaces.