Nonlinear partial differential equations (PDE’s) play a central role in the modelling of a great number of phenomena, ranging from Theoretical Physics, Astrophysics and Chemistry to Economy, Medicine and Population Dynamics. Among the phenomena encountered, the diffusion processes play a fundamental role. In the last 25 years a great deal of work has been devoted to semi-linear equations. In this type of equations, the interaction between a linear partial differential operator and the superlinear reaction term (source or absorption) can be understood, at least in part, thanks to the linear theory. One of the main observations, valid in the most interesting cases is the existence of critical exponents, for example the Fujita exponents for nonlinear heat equation, the Pohozaev exponent, the Sobolev exponent. In general, the first results (blow-up, global estimates, decay estimate), were proved up to a critical exponent by more or less easy applications of linear energy estimates, linked to ODE techniques. Then the study of what happens if the exponent is critical or even supercritical involves a very delicate analysis, often based upon very sharp linear estimates or even, in some cases a completely new approach. The main areas of research in the current proposal include the description of singular phenomenon: blow-up, singularities, problems with singular measure data in a large class of reaction diffusion equations, with quasi-linear or fully nonlinear diffusion and strong reaction.