In 1983 Gromov and Milman studies a topological fixed point theorem as an application of the theory of the L’evy-Milman concentration of measure phenomenon. They obtained that every continuous action of a Levy group on a compact metric space has a fixed point. Here a L’evy group is a metrizable topological group which is approximated from inside by an increasing chain of subgroups exhibiting the concentration of measure phenomenon. Nowadays many concrete examples of Levy groups are known (refer to the recent monograph by V. Pestov). Pursuing the idea of Gromov and Milman, we discuss actions of Levy groups on a large class of metric spaces from the viewpoint of the theory of concentration of maps.