Abstract: Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact non-compact group this is a non-metrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one point system (groups with the fixed point on compacta property), as well as topological groups $G$ for which M(G) is metrizable and the action of G on M(G) can be described explicitly. I will survey this new theory as developed by Glasner-Weiss, Pestov, Uspenskij, and Kechris-Pestov-Todorcevic, and show some connections with combinatorial Ramsey theory and the phenomenon of concentration of mass.