Free boundary problems are those that involve a priori unknown boundaries, whose location is part of the solution, and are ubiquitous in applications as they model a whole range of phenomena, from ocean-atmosphere interactions and flows in porous media to the propagation of fire fronts and option pricing of financial assets. Our goal as applied mathematicians is not only to understand the behavior of the weak solutions to the problems but also to provide information on the regularity and geometric properties of the pertaining interfaces.