Let $X$ denote the Gromov-Hausdorff limit of a noncollapsing sequence of riemannian manifolds $(M^n_i,g_i)$, with uniformly bounded Ricci curvature. Early workers conjectured (circa 1990) that $X$ is a smooth manifold off a closed subset of Hausdorff codimension $4$. We will explain a proof of this conjecture. This is joint work with Aaron Naber.