  We present a survey on several mass transportation problems, in which a given
mass dynamically moves from an initial configuration to a final one. The
approach we consider is the one introduced by Benamou and Brenier in [5], where
a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on
the velocity $v$ (which fulfill the continuity equation), has to be minimized.
Acting on the functional $F$ various forms of mass transportation problems can
be modeled, as for instance those presenting congestion effects, occurring in
traffic simulations and in crowd motions, or concentration effects, which give
rise to branched structures.
