We discuss the very basics of fully nonlinear elliptic equations, in close connection with regularity results. The latter include the celebrated Krylov-Safonov and Evans-Krylov theorems and the fundamental developments in Caffarelli’s theory. We also put forward more recent advances, such as the smoothness of flat solutions and the partial regularity result. In face of this panorama, we present some of our recent contributions to the theory. It covers fractional regularity estimates, the use of the Harnack approach and the connection with free boundary problems. We conclude with a discussion of a few open questions and their main challenges.