  Gauss periods taking exactly two values are closely related to two-weight
irreducible cyclic codes and strongly regular Cayley graphs. They have been
extensively studied in the work of Schmidt and White and others. In this paper,
we consider the question of when Gauss periods take exactly three rational
values. We obtain numerical necessary conditions for Gauss periods to take
exactly three rational values. We show that in certain cases, the necessary
conditions obtained are also sufficient. We give numerous examples where the
Gauss periods take exactly three vlaues. Furthermore, we discuss connections
between three-valued Gauss periods and combinatorial structures such as
circulant weighing matrices and 3-class association schemes.
