Geometric Measure Theory and Free Boundary Problems : Cetraro Italy 2019 This volume covers contemporary aspects of geometric measure theory with a focus on applications to partial differential equations free boundary problems and water waves. It is based on lectures given at the 2019 CIME summer school “Geometric Measure Theory and Applications – From Geometric Analysis to Free Boundary Problems” which took place in Cetraro Italy under the scientific direction of Matteo Focardi and Emanuele Spadaro. Providing an up to date account of the Yau conjecture a description of the structure of measures satisfying certain differential constraints and covering regularity theory for Bernoulli type free boundary problems and water waves it will be of interest to students and researchers in mathematical analysis and its applications.