Free boundary problems arise naturally in a number of physical phenomena and deal with solving partial differential equations in a domain, a part of whose boundary is unknown in advance. 

In this talk I will overview a famous free boundary problem: the obstacle problem. I will then focus on the thin obstacle problem for a divergence form elliptic operator, and I will describe a few methods used to tackle two fundamental questions: what is the optimal regularity of the solution, and what can be said about the geometry of the free boundary. 

The results are based on joint work with Nicola Garofalo and Arshak Petrosyan.