The aim of this paper is to provide new stability results for sequences of metric measure spaces (X i , d i , m i ) convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one (X, d), we extend the results of [GMS13] by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces H 1,p , including the space BV , and even with a variable exponent p i ∈ [1, ∞]. In addition, building on [AST16], we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for p > 1 and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case N < ∞, we improve some rigidity and almost rigidity results in [K15a, K15b, CM15a, CM15b]. On the basis of the second-order calculus in [G15b], in the class of RCD(K, ∞) spaces we provide stability results for Hessians and W 2,2 functions and we treat the stability of the Bakry-Émery condition BE(K, N ) and of Ric ≥ KI, with K and N not necessarily constant. * Scuola Normale Superiore, 11 Dimensional stability results 38 that in the class of smooth weighted n-dimensional Riemannian manifolds (M n , d, e −V vol M n ) the RCD * (K, N ) condition, n ≤ N , is equivalent toBy combining the continuity of (1.1) with the compactness property of the class of RCD * (K, N )-spaces w.r.t. the mGH convergence, we also establish a uniform boundwhere C i are positive constants depending only on K, N < ∞, and two-sided bounds of the diameter, i.e. C i do not depend on p (Proposition 11.1). Suspension theorems. The second application is related to almost spherical suspension theorems of positive Ricci curvature. For simplicity we discuss here only the case when N ≥ 2 is an integer, but our results (as those in [St06], [K15a], [K15b], [CM15b]) cover also the case N ∈ (1, ∞). In [CM15b] Cavalletti-Mondino proved that for any RCD * (N −1, N )space, the quantity (1.1) is greater than or equal than λ 1,p (S N , d, m N ) 1/p for any p ∈ [1, ∞), where S N is the unit sphere in R N +1 , d is the standard metric of the sectional curvature 1, and m N is the N -dimensional Hausdorff measure. Moreover, equality implies that the metric measure space is isomorphic to a spherical suspension. Under our notation (1.2) as above, this observation is also true when p = ∞, which corresponds to the Bonnet-Myers theorem in our setting (see [St06] by Sturm). Note that [CM15b] also provides rigidity results as the following one: for a fixed p ∈ [1, ∞], if (λ 1,p ) 1/p is close to λ 1,p (S N , d, m N ) 1/p , then the space is Gromov-Hausdorff close to the spherical suspension of a compact metric space, a so-called almost spherical suspension theorem. The converse is known for p ∈ {2, ∞} in [K15a, K15b] by Ketterer and we extend the result to general p; in addition, combining this with the joint spectral continuity result we can remove the p-dependence in the almost spherical suspension theore...