The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water, and one aims to describe the regularity of the interface separating the two phases.

In its stationary version, the Stefan problem reduces to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed, and that is constrained to lie above a given obstacle. 

The aim of this talk is to discuss some very recent developments on the optimal regularity of the free boundary both in the static and the parabolic setting.