Topological groups which are not locally compact feature quite prominently in various parts of mathematics. Examples are the unitary groups of Hilbert spaces, infinite symmetric groups, groups of isometries of non locally compact metric spaces (such as the Urysohn space), groups of homeomorphisms, and so forth.

The properties of such groups are, predictably, not the same as of their locally compact counterparts, and quite often in the locally compact case they are either trivially true or trivially false. In this small survey we will try to paint the picture of the present state of the study of such groups, putting an emphasis on concrete examples. In tune with the topic of the thematic program, we will pay a special attention to the role played by concentration of measure, (finite) oscillation stability, and Ramsey theory.