This paper introduces a notion of nonnegative N-Ricci curvature and lower bounded ∞-Ricci curvature for measured length spaces using optimal transport and displacement convexity. The authors define these properties in terms of the convexity of certain functions on the Wasserstein space of probability measures on the space. They show that these properties are preserved under measured Gromov–Hausdorff limits and derive geometric and analytic consequences. The paper has two main goals: extending results from smooth Riemannian manifolds to length spaces and using optimal transport to define a notion of Ricci curvature for measured length spaces. The authors define a measured length space (X, d, ν) to have nonnegative N-Ricci curvature if certain functions are convex along Wasserstein geodesics. They also define ∞-Ricci curvature bounded below by K if similar conditions hold for a different class of functions. The authors show that these definitions are equivalent to classical notions of Ricci curvature for Riemannian manifolds. They also prove that measured Gromov–Hausdorff limits of manifolds with lower Ricci curvature bounds fall under their considerations. Additionally, they derive analytic consequences, such as log Sobolev inequalities, from these definitions. The paper is structured into sections that cover basic definitions, the geometry of the Wasserstein space, displacement convexity, and the main results. Appendices provide additional technical results and discussions on optimal transport and displacement convexity. The authors also mention related work by other researchers and thank the anonymous referees for their suggestions.This paper introduces a notion of nonnegative N-Ricci curvature and lower bounded ∞-Ricci curvature for measured length spaces using optimal transport and displacement convexity. The authors define these properties in terms of the convexity of certain functions on the Wasserstein space of probability measures on the space. They show that these properties are preserved under measured Gromov–Hausdorff limits and derive geometric and analytic consequences. The paper has two main goals: extending results from smooth Riemannian manifolds to length spaces and using optimal transport to define a notion of Ricci curvature for measured length spaces. The authors define a measured length space (X, d, ν) to have nonnegative N-Ricci curvature if certain functions are convex along Wasserstein geodesics. They also define ∞-Ricci curvature bounded below by K if similar conditions hold for a different class of functions. The authors show that these definitions are equivalent to classical notions of Ricci curvature for Riemannian manifolds. They also prove that measured Gromov–Hausdorff limits of manifolds with lower Ricci curvature bounds fall under their considerations. Additionally, they derive analytic consequences, such as log Sobolev inequalities, from these definitions. The paper is structured into sections that cover basic definitions, the geometry of the Wasserstein space, displacement convexity, and the main results. Appendices provide additional technical results and discussions on optimal transport and displacement convexity. The authors also mention related work by other researchers and thank the anonymous referees for their suggestions.