Abstract We study amenability of definable groups and topological groups, prove various results, briefly described below. Among our main technical tools, interest in its own right, is an elaboration on strengthening the Massicot-Wagner version (Massicot Wagner J Ec Polytech Math 2:55–63, 2015) stabilizer theorem (Hrushovski Am Soc 25:189–243, 2012), also some results about measures measure-like...