  Free boundary problems are those described by PDEs that exhibit a priori
unknown (free) interfaces or boundaries. These problems appear in Physics,
Probability, Biology, Finance, or Industry, and the study of solutions and free
boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure
Theory, and Harmonic Analysis. The most important mathematical challenge in
this context is to understand the structure and regularity of free boundaries.
  In this paper we provide an invitation to this area of research by
presenting, in a completely non-technical manner, some classical results as
well as some recent results of the author.
