The story how analysis met geometric measure theory and what they were doing together?

Starting with the early 1990's several outstanding problems of complex analysis were solved by new methods that involved non-homogeneous harmonic analysis, geometric measure theory and random geometric constructions. I mean here the problems of Painlev\'e, Ahlfors, Vitushkin, Denjoy and solved by Tolsa, David--Mattila and Nazarov--Treil--Volberg. This lucky encounter continued for the next 20 years, where those methods were used to solve several free boundary problems of Bishop and Guy David. I will try to give the exposition of ideas behind this development.