Abstract:  Let M be a manifold with bounded Ricci curvature and which is noncollapsed, i.e. |Rc|v>0.  Together with Wenshuai Jiang we prove that there then exists the apriori L^2 curvature estimate \int |Rm|^2 < C and L^1 hessian estimate \int |\nabla^2 Rm| < C.  In particular we conclude that the singular set of a non smooth limit has finite n-4 measure, which resolves a conjecture of Cheeger/Colding.  The proofs involve several new ingredients, including a new log superconvexity estimate for harmonic functions, which is the key ingredient to 'decay growth' type estimates for non isolated singularities.