A Survey on dynamical transport distances

In this paper we review some transport models based on the continuity equation, starting with the so-called {\it Benamou-Brenier formula}, which is nothing but a fluid mechanics reformulation of the Monge-Kantorovich problem with cost $c(x,y)=|x-y|^2$. We discuss some of its applications (gradient flows, sharp functional inequalities...), as well as some variants and generalizations to dynamical transport problems, where interaction effects among mass particles are considered