Abstract: The Lott-Sturm-Villani theory of CD(K, N) metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport, though extremely successful, has been shown not to directly apply to sub-Riemannian geometries. Nonetheless, still using optimal transport tools, some entropy inequalities have been proved to hold in the case of the Heisenberg group and more in general in sub-Riemannian manifolds. In this talk we survey the known results and motivate a new approach we propose aiming to unify Riemannian and sub-Riemannian synthetic Ricci lower bounds, introducing suitable curvature dimension conditions. (joint with Andrea Mondino (Oxford) and Luca Rizzi (SISSA)) This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.