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).