The phenomenon of measure concentration, in its modern formulation, was isolated in the late 1960s and early 1970s by Vitali Milman, extending an idea going back to Paul Levy's work on the geometry of Euclidean spheres, and has since led to numerous interesting applications in geometry, analysis, and combinatorics. Roughly speaking, concentration of measure means that, on a high-dimensional space, every uniformly continuous function is close to being constant everywhere except on a set of a small measure. In their groundbreaking 1983 joint work, Misha Gromov and Vitali Milman linked this phenomenon with topological dynamics, by proving a universal fixed point theorem for continuous actions of a certain type of topological groups, so-called Levy groups. Examples of such topological groups include the unitary group of the infinite-dimensional separable Hilbert space equipped with the strong operator topology, the isometry group of the Urysohn metric space with the topology of point-wise convergence, and the automorphism group of the non-atomic standard Lebesgue space with the weak topology. In my talk, I will give an overview of measure concentration and its most prominent manifestations, explain its connection with topological dynamics and ergodic theory, and sketch some recent developments concerning Gromov's idea of concentration to non-trivial spaces.