This chapter is broadly divided into two parts. In the first, a brief but self-contained review of the interior regularity theory of elliptic PDEs is presented, including relevant preliminaries on function spaces. In the second, as an application of the tools introduced in the first part, a detailed study of the Einstein condition on Riemannian manifolds with metrics of Holder regularity is undertaken, introducing important techniques such as the use of harmonic coordinates and giving some consideration to the smoothness of the differentiable structure of the underlying manifold.