For a topological group G, let Aut(G) be the group of all automorphisms, and let w(G) be the number of orbits of Aut(G) on G. I will report about results characterizing classes of groups G with suitable (topological or algebraic) assumptions on G, combined with bounds on w(G). In particular, there are strong results in the following cases: * G is locally compact and connected, with w(G) less than the cardinality of the continuum * G is compact, and w(G) is finite * G is finite, and w(G) < 5