Regularity of free boundaries for obstacle type problems

The approach based on monotonicity and non-degeneracy to study the regularity of free boundaries will be discussed. In particular recent results obtained jointly with H. Shahgholian and G. Weiss will be presented. For two-phase obstacle problem it is proved that if two phases are close enough to each other at the origin then in some ball Br(0) their boundaries are C1 graphs.