Abstract : After a short review on Nonlinear and Fractional Diffusion, I will report on two recent issues. One deals with the theory of the fractional p-Laplacian equation in the whole space, which is the gradient flow for the Gagliardo energy. Main news are the construction of selfsimilar solutions in R^n, that attract all finite-mass solutions. I also report on how the Gagliardo fractional seminorm comes naturally in estimating Taylor expansions and other topics of appoximation theory.