  We discuss a connection between two areas of mathematics which until recently
seemed to be rather distant from each other: (1) noncommutative harmonic
analysis on groups and (2) some topics in probability theory related to random
point processes. In order to make the paper accessible to readers not familiar
with either of these areas, we explain all needed basic concepts.
  This is an extended version of G.Olshanski's talk at the 4th European
Congress of Mathematics.
