The dynamics for the interface in many free boundary problems is driven by the normal derivative of a corresponding pressure function, and often the free boundary is the boundary of the positivity set of this unknown pressure, which evolves in time. One well-known example of these types of models is called the Hele-Shaw problem. In this talk we will describe the set-up of some of these free boundaries and show how in many situations, their solution becomes equivalent to solving a nonlinear fractional heat equation (in one fewer space variables). These fractional heat equations fall into the general scope of what are called Hamilton-Jacobi-Bellman equations, which have enoyed extensive study in the past 20 years or so (at least for the fractional setting, and much longer for the first and second order settings). Furthermore, many of the well established properties about existence, uniqueness, and regularity for Hamilton-Jacobi-Bellman equations can then be transferred back to the original free boundary problem. We will discuss various recent results in this direction.