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For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannianmanifolds with a uniform bound of Ricci curvature, we establish two spectralconvergence. One of them is on the Hodge Laplacian acting on differentialone-forms. The other is on the connection Laplacian acting on tensor fields ofevery type, which include all differential forms. These are generalizations ofCheeger-Colding's spectral convergence of the Laplacian acting on functions tothe cases of tensor fields and differential forms. As a corollary we show theupper semicontinuity of the first Betti numbers with respect to theGromov-Hausdorff topology, and give the equivalence between the continuity ofthem and the existence of a uniform spectral gap. We also define the Riemanniancurvature tensor, the Ricci curvature, and the scalar curvature of the limitspace, and show their properties. In particular the Ricci curvature coincideswith the difference between the Hodge Laplacian and the connection Laplacian,and is compatible with Gigli's one and Lott's Ricci measure. Moreover a lowerbound of the Ricci curvature is compatible with a reduced Riemannian curvaturedimension condition. We also give a positive answer to Lott's question on thebehavior of the scalar curvature with respect to the Gromov-Hausdorff topology.

Spectral convergence under bounded Ricci curvature