Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact non-compact group this is a nonmetrizable 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 (extremely amenable groups), as well as topological groups G for which M(G) is metrizable (in fact Cantor) space and for which one has an explicit description. I will survey this new theory as developed by Pestov, Uspenskij, Glasner and Weiss and Kechris, Pestov, and Todorcevic, and show how it relies on combinatorial Ramsey type theorems.