We discuss finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of numerical schemes. Our focus is on a pure equation and a related free boundary problem of transmission type. The cornerstone of our argument is a regularisation procedure. It decouples the degeneracy term from the elliptic operator driving the diffusion process. In the free boundary setting, the absence of degenerate ellipticity entails new, genuine difficulties. To bypass them, we resort to the intrinsic properties of the regularised problem. We conclude with numerical experiments and an overview of future research directions. This is joint work with Ercilia Sousa (CMUC and Universidade de Coimbra).