Elementary spherical functions on semisimple Lie groups suggest by “variation of structural parameters” the concept of hypergeometric functions associated with root systems. Such a theory was developped several years ago by Opdam and the lecturer. Dunkl operators are a mixture of differential and divided difference operators, and they form a basic tool in the differential algebra of this hypergeometric theory. It is remarkable to see the ubiquitous role played by the representation theory of affine Hecke algebras and their degenerations