The paper introduces the concept of lower Ricci curvature bounds for metric measure spaces, which generalizes the notion of lower curvature bounds for Riemannian manifolds introduced by A.D. Alexandrov. The main contributions include: 1. **Definition of a Metric Measure Space**: A metric measure space is defined as a triple \((M, \mathbf{d}, m)\) where \((M, \mathbf{d})\) is a complete and separable metric space, and \(m\) is a locally finite measure on the Borel \(\sigma\)-algebra of \(M\). 2. **Complete and Separable Metric \(\mathbf{D}\)**: A complete and separable metric \(\mathbf{D}\) is defined on the family of isomorphism classes of normalized metric measure spaces. This metric is based on the optimal mass transportation problem and has a natural interpretation in terms of the \(L_2\)-Wasserstein distance. 3. **D-convergence**: D-convergence is shown to be weaker than measured Gromov–Hausdorff convergence and is equivalent to it for compact metric measure spaces with full supports and uniform bounds for the doubling constant and diameter. 4. **Lower Curvature Bounds**: Lower curvature bounds \(\mathbf{Curv}(M, \mathbf{d}, m)\) are introduced for metric measure spaces, based on convexity properties of the relative entropy \(\mathrm{Ent}(\cdot|m)\) with respect to the reference measure \(m\). These bounds are stable under D-convergence and imply estimates for the volume growth of concentric balls. 5. **Local to Global Lower Curvature Bounds**: Local lower curvature bounds are shown to imply global lower curvature bounds. 6. **Compactness Results**: The family of normalized metric measure spaces with curvature \(\geq K\), doubling constant \(\leq C\), and diameter \(\leq L\) is shown to be compact under D-convergence. 7. **Applications**: Lower curvature bounds imply various geometric and analytic properties, such as estimates for the volume growth, the Bishop–Gromov volume comparison theorem, and the Bonnet–Myers theorem. The paper also discusses the geometric interpretation of the \(L_2\)-Wasserstein distance and its role in the theory, as well as the relationship between curvature bounds and the heat semigroup on metric measure spaces.The paper introduces the concept of lower Ricci curvature bounds for metric measure spaces, which generalizes the notion of lower curvature bounds for Riemannian manifolds introduced by A.D. Alexandrov. The main contributions include: 1. **Definition of a Metric Measure Space**: A metric measure space is defined as a triple \((M, \mathbf{d}, m)\) where \((M, \mathbf{d})\) is a complete and separable metric space, and \(m\) is a locally finite measure on the Borel \(\sigma\)-algebra of \(M\). 2. **Complete and Separable Metric \(\mathbf{D}\)**: A complete and separable metric \(\mathbf{D}\) is defined on the family of isomorphism classes of normalized metric measure spaces. This metric is based on the optimal mass transportation problem and has a natural interpretation in terms of the \(L_2\)-Wasserstein distance. 3. **D-convergence**: D-convergence is shown to be weaker than measured Gromov–Hausdorff convergence and is equivalent to it for compact metric measure spaces with full supports and uniform bounds for the doubling constant and diameter. 4. **Lower Curvature Bounds**: Lower curvature bounds \(\mathbf{Curv}(M, \mathbf{d}, m)\) are introduced for metric measure spaces, based on convexity properties of the relative entropy \(\mathrm{Ent}(\cdot|m)\) with respect to the reference measure \(m\). These bounds are stable under D-convergence and imply estimates for the volume growth of concentric balls. 5. **Local to Global Lower Curvature Bounds**: Local lower curvature bounds are shown to imply global lower curvature bounds. 6. **Compactness Results**: The family of normalized metric measure spaces with curvature \(\geq K\), doubling constant \(\leq C\), and diameter \(\leq L\) is shown to be compact under D-convergence. 7. **Applications**: Lower curvature bounds imply various geometric and analytic properties, such as estimates for the volume growth, the Bishop–Gromov volume comparison theorem, and the Bonnet–Myers theorem. The paper also discusses the geometric interpretation of the \(L_2\)-Wasserstein distance and its role in the theory, as well as the relationship between curvature bounds and the heat semigroup on metric measure spaces.