Long range or anomalous diffusions, such as diffusions given by the fractional powers $(-\Delta)^s$ of the Laplacian, attract lately interest in Physics, Biology, and Finance. From the mathematical point of view, nonlinear analysis for fractional diffusions is being developed actively in the last years. In this talk, I will describe recent results concerning front propagation for the nonlinear fractional KPP heat equation, $\partial_t tu+(-\Delta)^su = u(1-u) in (0;1) Rn, 0   u   1$, with $s 2 (0; 1)$. In collaboration with J. M. Roquejore, we establish that fronts propagate at exponential speed |in contrast with the classical case $s=1$ for which there is propagation at a constant KPP speed. I will also describe works in collaboration with Y. Sire and E. Cinti. They concern the fractional elliptic Allen-Cahn equation $(-\Delta)^su = f(u)$ in $Rn$ with $s 2 (0; 1)$, a model being the bistable nonlinearity $f(u) = u - u3$. Our main results concern the existence and properties of layer or heteroclinic solutions, as well as of minimizers of the equation.