  In this survey we stress the importance of the higher transcendental
Mittag-Leffler function in the framework of the Fractional Calculus. We first
start with the analytical properties of the classical Mittag-Leffler function
as derived from being the solution of the simplest fractional differential
equation governing relaxation processes. Through the Sections of the text we
plan to address the reader in this pathway towards the main applications of the
Mittag-Leffler function that has induced us in the past to define it as the
Queen Function of the Fractional Calculus.
  These applications concern some noteworthy stochastic processes and the time
fractional diffusion-wave equation. We expect that in the next future this
function will gain more credit in the science of complex systems.
  In Appendix A we sketch some historical aspects related to the author's
acquaintance with this function. Finally, with respect to the published version
in Entropy, we add Appendix B where we briefly refer to the numerical methods
nowadays available to compute the functions of the Mittag-Leffler type.
