Our goal is to accomplish a leap forward in the knowledge on propagation phenomena in reaction-diffusion equations, in heterogeneous media and/or non standard diffusion, systems as well as non local interactions. This proposal deals both with the general theory of nonlinear PDE’s of elliptic and parabolic type as well as with the development and study of some specific models. These range from ecology, medicine, mathematical economics and social sciences.Reaction-diffusion models, especially in ecology (for instance those describing biological invasions), feature long range interactions and heterogeneities, whose understanding is a current outstanding challenge. Models in theoretical medicine couple multi-scale phenomena to complex geometries and mixtures of local and nonlocal interactions. Economy is a constant source of newand nonstandard free boundary problems. We therefore propose to bring our expertise in propagation phenomena for reaction-diffusion, calculus of variations and free boundary problems, to treat a large class of these new models. The level of both generality and precision we are aiming at has not, to our knowledge, been reached before.The project is especially timely: on the one hand, the international activity in reaction-diffusion equations and all related topics is intense. On the other hand, the modelling activity in theoretical biology, ecology, medicine and social sciences is experiencing a considerable growth. The PI of this proposal being at the leading edge of both fields, there is a unique occasion to give a new impulse to a domain that is important both to mathematical analysis and to its potential applications.