This talk will concern joint work with Aaron Naber on the regularity of noncollapsed Riemannian manifolds $M^n$ with bounded Ricci curvature $|{\rm Ric}_{M^n}|\leq n-1$, as well as their Gromov-Hausdorff limit spaces, $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow}(X,d)$, where $d_j$ denotes the Riemannian distance. We will explain a proof of the conjecture that $X$ is smooth away from a closed subset of  codimension $4$. By combining this with the ideas of quantitative stratification, we obtain a priori $L^q$ estimates on the full curvature tensor, for all $q0$, $