**Title:** Regularity for nonlocal parabolic equations

**Abstract:** Nonlocal (or fractional) differential equations can be effective in modeling phenomena in the physical sciences such as signals, evolution, percolation, and thermodynamics. In this talk we will discuss a recent parabolic fractional equation which models flow in plasma transport. We will discuss existence, uniqueness, and regularity for this equation. We will show how the nonlocal nature of the equation can be utilized in proving these properties. We will also demonstrate how these techniques can be adapted to prove regularity for a nonlinear fractional porous medium type equation. This is joint work with L. Caffarelli and A. Vasseur.