  Selberg sums are the analogues over finite fields of certain integrals
studied by Selberg in in 1940s. The original versions of these sums were
introduced by R.J.Evans in 1981 and, following an elegant idea of G.W.Anderson
in 1991 they were evaluated by Anderson, Evans and P.B.~van~Wamelen. In 2007
the author noted that these sums and certain generalizations of them appear in
the study of the distribution of Gauss sums over a rational function field over
a finite field. The distribution of Gauss sums is closely related to the
distribution of the values of the discriminant of polynomials of a fixed
degree. Here we shall take this up further. The main goal here is to establish
the basic properties of Selberg sums and to formulate the problems which arise
from this point of view.
