This chapter contains an exposition on the theory of (finite) Gelfand pairs and their spherical functions. This theory originally was developed in the setting of Lie groups with the seminal paper by I. M. Gelfand (see also) another earlier contribution is the paper by Godement.

Expositions of the theory in the setting of locally compact and/or Lie groups are in Dieudonnè's treatise on analysis and in the monographs by: Dym and McKean, Faraut, Figà-Talamanca and Nebbia, Helgason, Klimyk and Vilenkin, Lang and Ricci. See also the papers by Bougerol.

Recently, finite and infinite Gelfand pairs have been studied in asymptotic and geometric group theory in connection with the so-called branch groups introduced by R. I. Grigorchuk in (see).

Several examples of finite Gelfand pairs, where G is a Weyl group or a Chevalley group over a finite field were studied by Delsarte, Dunkl and Stanton (see the surveys or the book by Klimyk and Vilenkin). Delsarte was motivated by applications to association schemes of coding theory, while Dunkl and Stanton were interested in applications to orthogonal polynomials and special functions. For the point of view of the theory of association schemes see the monographs by Bailey, Bannai and Ito see also the work of Takacs, on harmonic analysis on Schur algebras, that contains several applications to probability.