  Dunkl theory is a far reaching generalization of Fourier analysis and special
function theory related to root systems. During the sixties and seventies, it
became gradually clear that radial Fourier analysis on rank one symmetric
spaces was closely connected with certain classes of special functions in one
variable. During the eighties, several attempts were made, mainly by the Dutch
school, to extend these results in higher rank (i.e. in several variables),
until the discovery of Dunkl operators in the rational case and Cherednik
operators in the trigonometric case. Together with q-special functions
introduced by Macdonald, this has led to a beautiful theory, developed by
several authors, which encompasses in a unified way harmonic analysis on all
Riemannian symmetric spaces and spherical functions thereon.In this series of
lectures, delivered at the Summer School AAGADE 2015 (Analytic, Algebraic and
Geometric Aspects of Differential Equations, Mathematical Research and
Conference Center, Bedlewo, Poland, September 2015), we aim at giving an
updated overview of Dunkl theory, with an emphasis on its analytic aspects.
