Geometric measure theory and the study of free boundary problems in PDE share many methods and techniques. Indeed, the one phase free boundary problem shares a well-known connection to area-minimizing surfaces. In this talk we review this connection and explain how many results for area minimizing surfaces have analogous results for the one-phase free boundary problem. I will then discuss recent results for the one-phase problem on cones. After reviewing results of the author with H. Chang Lara on two-dimensional cones, we revisit the connection to area-minimizing surfaces to gain insight into the problem on higher dimensional cones. We then present new results on when the free boundary is allowed to pass through the vertex of a three-dimensional cone as well as results for higher dimensional cones.