  In this review paper we stress the importance of the higher transcendental
Wright functions of the second kind in the framework of Mathematical Physics.We
first start with the analytical properties of the classical Wright functions of
which we distinguish two kinds. We then justify the relevance of the Wright
functions of the second kind as fundamental solutions of the time-fractional
diffusion-wave equations. Indeed, we think that this approach is the most
accessible point of view for describing non-Gaussian stochastic processes and
the transition from sub-diffusion processes to wave propagation. Through the
sections of the text and suitable appendices we plan to address the reader in
this pathway towards the applications of the Wright functions of the second
kind. Keywords: Fractional Calculus, Wright Functions, Green's Functions,
Diffusion-Wave Equation,
