Fractional diffusion is the most widespread application of fractional calculus. Actually, it gave to fractional calculus the required visibility for becoming a remarkable tool in modelling, because of explaining unexpected experimental results, a fascinating topic in analysis, because of challenging established theorems, and a stimulating subject in probability, because of its relation with power-law Lévy stable densities. Roughly speaking, fractional equations are generalised equations where integer-order derivatives are replaced by their real-order counterparts. In the seminar, the advancements on the study of fractional diffusion will be retraced with the results obtained by the speaker. Starting from the derivation for the first time of the exact solutions of the space-time fractional diffusion equation in his Laurea thesis [1], passing through the origin of fractional diffusion from heterogeneity in standard Gaussian processes [2,3], to end with the last paper where the convergence of Lévy-type random walks to the solution of the fractional diffusion equation is shown to be indeed not always guaranteed [4]. The significance of this last result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to recurrence and the related concept of site fidelity in the framework of Lévy-like motion for wild animals. [1] Mainardi F., Luchko Yu., Pagnini G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 4, 153-192 (2001) [2] Molina-García D., Pham T. Minh, Paradisi P., Manzo C., Pagnini G., Fractional kinetics emerging from ergodicity breaking in random media. Phys. Rev. E. 94, 052147 (2016) [3] D’Ovidio M., Vitali S., Sposini V., Sliusarenko O., Paradisi P., Castellani G., Pagnini G., Centre-of- mass like superposition of Ornstein-Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion. Fract. Calc. Appl. Anal. 21, 1420-1435 (2018) [4] Pagnini G., Vitali S., Should I stay or should I go? Zero-size jumps in random walks for Lévy flights. Fract. Calc. Appl. Anal. 24, 137-167 (2021)