This paper introduces a new concept of lower Ricci curvature bounds for metric measure spaces, which are triples (M, d, m) where (M, d) is a metric space and m is a measure on the Borel σ-algebra of M. The main results include the definition of a complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces, which has a natural interpretation based on optimal mass transportation. D-convergence is weaker than measured Gromov-Hausdorff convergence, and both are equivalent on families of compact metric measure spaces with certain properties. The paper also introduces a notion of lower curvature bounds for metric measure spaces based on convexity properties of the relative entropy with respect to the reference measure m. These bounds are stable under D-convergence and imply volume growth estimates for concentric balls. The paper also discusses the relationship between curvature bounds and functional inequalities like Sobolev and Poincaré inequalities. The concept of optimal mass transportation plays a crucial role in the approach, with the L₂-Wasserstein distance being a key tool. The paper also discusses the relationship between curvature bounds and the geometry of metric measure spaces, including the curvature-dimension condition CD(K, N), which is more restrictive than the previous condition. The paper concludes with a discussion of the implications of these results for the geometry of metric measure spaces and the stability of curvature bounds under convergence.This paper introduces a new concept of lower Ricci curvature bounds for metric measure spaces, which are triples (M, d, m) where (M, d) is a metric space and m is a measure on the Borel σ-algebra of M. The main results include the definition of a complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces, which has a natural interpretation based on optimal mass transportation. D-convergence is weaker than measured Gromov-Hausdorff convergence, and both are equivalent on families of compact metric measure spaces with certain properties. The paper also introduces a notion of lower curvature bounds for metric measure spaces based on convexity properties of the relative entropy with respect to the reference measure m. These bounds are stable under D-convergence and imply volume growth estimates for concentric balls. The paper also discusses the relationship between curvature bounds and functional inequalities like Sobolev and Poincaré inequalities. The concept of optimal mass transportation plays a crucial role in the approach, with the L₂-Wasserstein distance being a key tool. The paper also discusses the relationship between curvature bounds and the geometry of metric measure spaces, including the curvature-dimension condition CD(K, N), which is more restrictive than the previous condition. The paper concludes with a discussion of the implications of these results for the geometry of metric measure spaces and the stability of curvature bounds under convergence.