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In this survey we stress the importance of the higher transcendentalMittag-Leffler function in the framework of the Fractional Calculus. We firststart with the analytical properties of the classical Mittag-Leffler functionas derived from being the solution of the simplest fractional differentialequation governing relaxation processes. Through the Sections of the text weplan to address the reader in this pathway towards the main applications of theMittag-Leffler function that has induced us in the past to define it as theQueen Function of the Fractional Calculus.  These applications concern some noteworthy stochastic processes and the timefractional diffusion-wave equation. We expect that in the next future thisfunction will gain more credit in the science of complex systems.  In Appendix A we sketch some historical aspects related to the author'sacquaintance with this function. Finally, with respect to the published versionin Entropy, we add Appendix B where we briefly refer to the numerical methodsnowadays available to compute the functions of the Mittag-Leffler type.

Why the Mittag-Leffler function can be considered the Queen function of the Fractional Calculus?