The theory of minimal surfaces and the study of free boundary problems (FBP) are intimately related. In this talk I will discuss a couple of the deep and fruitful connections between the two subjects. I will then present joint work with David Jerison, in which we describe the space of solutions to the one-phase FBP in the disk, having simply-connected positive phase. Our results are direct counterparts to theorems by Colding and Minicozzi for minimal surfaces.