  In this tutorial survey we recall the basic properties of the special
function of the Mittag-Leffler and Wright type that are known to be relevant in
processes dealt with the fractional calculus. We outline the major applications
of these functions. For the Mittag-Leffler functions we analyze the Abel
integral equation of the second kind and the fractional relaxation and
oscillation phenomena. For the Wright functions we distinguish them in two
kinds. We mainly stress the relevance of the Wright functions of the second
kind in probability theory with particular regard to the so-called M-Wright
function that generalizes the Gaussian and is related with the time-fractional
diffusion equation.
