Optimal transport is an old branch of the calculus of variations whose origins can be traced back to an important memoir of Monge in 1781, followed by remarkable contributions due to Kantorovich in 1942, and in the last 50 years by R.L. Dobrushin, Y. Brenier, and many others. Among the by-products of optimal transport is a family of distances metrizing the weak topology of Borel probability measures on Euclidean spaces. The analogy between Borel probability measures on phase space and the notion of density operators used in quantum mechanics suggests defining a notion of « pseudometric » which can be used to compare two (quantum) density operators, or a density operator with a probability density in phase space. The talk will discuss the main properties of this pseudometric, and applications to some problems arising in quantum dynamics (such as the classical limit, or the mean-field limit of large particle systems…) This presentation is based on a series of works with E. Caglioti, C. Mouhot and T. Paul.