Differential inclusions have proved to be an extensively valuable tool in studying PDEs, with applications ranging from establishing the existence of solutions to constrained variational problems to constructing pathological examples in various classes of problems. The talk aims to briefly review the theory of solvability for differential inclusions outlining the different approaches that can be used and discuss some recent applications. One of the characteristics of the theory is that, when applicable, it generates infinitely many solutions; therefore, the problem of finding a criterion to select among them the more meaningful ones naturally arises. Although there is still no general method to address this problem, we will see that a tailored selection criterion can be successfully applied for some problems.