A weakly continuous near-action of a Polish group G on a standard Lebesgue measure space (X,µ) is whirly, or ergodic at identity, if for every A ⊆ X of strictly positive measure and every neighbourhood V of identity in G the set V A has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic action by a Polish L´evy group in the sense of Gromov and Milman, such as U(ℓ 2 ), is whirly (Glasner–Tsirelson–Weiss). We survey this direction of research, and in particular give examples of closed subgroups of the group Aut(X,µ) of measure preserving automorphisms of a standard Lebesgue measure space (with the weak topology) whose tautological action on (X,µ) is whirly, and which are not L´evy groups, thus answering a question of Glasner and Weiss. Some open questions are discussed.