Generalizing classical work of Day and Følner for discrete groups, I will present characterizations of amenability of (not necessarily locally compact) topological groups in terms of the existence of almost invariant vectors and almost invariant finite subsets, and discuss some applications of these results, e.g., concerning the coarse geometry of Polish groups. This is joint work with Andreas Thom. Furthermore, linked with concentration of measure, the mentioned amenability criteria provide sufficient conditions for (a strong form of) extreme amenability, which can be used to prove the extreme amenability of topological groups of measurable maps. This is joint work with Vladimir Pestov.