  This small survey of basic universal constructions related to the actions of
topological groups on compacta is centred around a new result --- an intrinsic
description of extremely amenable topological groups (i.e., those having a
fixed point in each compactum they act upon), solving a 1967 problem by
Granirer. Another old problem whose solution (in the negative) is noted here,
is a 1957 conjecture by Teleman on irreducible representations of general
topological groups. Our exposition covers the greatest ambit and the universal
minimal flow, as well as closely related constructions and results from
representation theory. We present in a simplified fashion the known examples of
extremely amenable groups and discuss their relationship with a 1969 problem by
Ellis. Open questions, including those from bordering disciplines, are reviewed
along the way.
