VALDINOCI Nonlocal nonlinear problems Abstract: I would like to discuss some questions related to some semilinear equations driven by a nonlocal elliptic operator (for example, the Allen-Cahn equation, in which the classical Laplace operator is replaced by a fractional Laplacian). In particular, I will study the qualitative properties of the solutions, such as symmetry, density estimates of the level sets, asymptotic behaviors, etc. The limit interfaces of these problems are related to both the local and the nonlocal perimeter functionals: on this topic, I will discuss some rigidity and regularity results and to present some open problems.