Given a Riemannian manifold (that is a particular metric space on which the notion volume can be defined) a very natural question is to understand how the volume of balls varies depending on its radius and/or its center. Formalizing this idea of the interplay between distances and volumes led to a rich theory known as “theory of Riemannian manifolds with Ricci curvature bounded below”. Rather than focusing on technical details, I will illustrate the power of this theory with numerous examples. On the other hand one can see a volume element as a distribution of mass that one wants to transport (reorganize) to an other volume element (distribution of mass). Given a cost of moving volume elements, one might wonder what is the most cost-effective way of transporting the mass. This problem is formally known as “optimal transport”. When the cost depends explicitly on the distance, optimal transport strongly intertwines volumes and distances. The volume-distance intrication appearing both in optimal transport and the theory of Riemannian manifolds with Ricci curvature bounded below suggests a strong link between the two theories. In the last 15 years, this link has been discovered and explicated : any statement of the theory of Ricci curvature bounded below can be rephrased in an optimal transport fashion. The optimal transport point of view does not require any smoothness or regularity assumptions and can therefore be used to study rougher spaces.