introduced 13 years ago, daha play now a solid role in modern representation theory with wide spectrum of applications from harmonic analysis and combinatorics to algebraic geometry and mathematical physics. i will begin with general motivation in the theory of spherical functions and the definitions in the rank one case. the talk will be mainly about the so-called nonsymmetric verlinde algebras, which are in the focus of the new theory, including applications to the ''non-cyclotomic'' gaussian sums and recent results on the diagonal coinvariants (the combinatorics of two sets of variables).