In many problems in fluid dynamics, the fluid has a solid boundary on only some sides, with a free surface forming the rest of the boundary (the top surface of the ocean is an example of such a free surface). Thus, one of the unknowns when studying initial value problems is the domain of the functions being considered. In this talk, I will discuss my formulations of some such free-surface problems in two-dimensional irrotational fluids. The formulation primarily involves choosing convenient dependent variables and a parameterization for the free surface. After the problems are formulated in this way, it can be shown using energy methods that the initial value problem is well-posed (i.e, that solutions exist, are unique, and depend continuously on the initial data). A similar formulation can be made in the case of three-dimensional fluids; I will discuss this if time permits.