The development of amenability in the past 80 years

The notion of an amenable group (under a different name) was introduced by von Neumann in 1929 and had became quite popular in different areas of mathematics: functional analysis, dynamical systems, operator algebras, theory of probability, and other. The examples of amenable groups that were known to von Neumann were limited to the so called elementary amenable groups.
In 1957 M. Day asked if there exist nonelementary amenabe groups. First such groups were found by the speaker in 1983 as a consequence of the solution of Milnor Problem on groups of intermediate growth.
In 1987 I posed the question on the existence of amenable but not subexponentially amenable groups, which was solved recently by L. Bartholdi and B. Virag, who used some preliminary results of the speaker and his collaborators. The group that leads to the solution of the problem is called Basilica. It played a fundamental role in the creation of the itereted monodromy group theory by V. Nekrashevych – a new powerful tool in holomorphic dynamics.
We will speak about the solution of the problem on nonsubexponential amenability, based on the theory of self-similar groups of fractal type, and about the method used to prove the amenability, which was named the “Munchhausen trick” (and which has roots in Kesten's probabilistic criterion of amenability and entropy theory of random walks).
At the end of the talk new open problems will be posed and some interesting classes of groups will be discussed.