A random point process is a random variable taking values in the space of discrete countable subsets of a space X. In other words, given a point process in X, one can speak about random point configurations in X. The best know example is the Poisson process on the real line, which is widely used in queueing theory. Another important class of examples consists of the so-called determinantal point processes. These point processes were mainly studied in connection with asymptotic problems in random matrix theory. Here point configurations are obtained from matrix eigenvalues. The aim of the talk is to describe (avoiding technical details) how determinantal point processes arise in a different part of mathematics – harmonic analysis on noncommutative groups. No special prerequisites are required; the necessary concepts will be explained.