The classical Monge problem (but only with cost $|x-y|$, and not $|x-y|^2$) has an equivalent counterpart which is the minimization of the $L^1$ norm of a vector field $v$, subject to the constraint $\nabla\cdot v=\mu-\nu$. This is a minimal flow problem introduced by M. Beckmann in the '50s without knowing the relation with the works by Kantorovich. It has recently come back into fashion because of its possible variants, where the cost rather than being linear as in the $L^1$ norm can be made convex (thus taking into account for congestion effects) or concave (favoring joint transportation). Also, well-posedness of this problem and regularity issues about the optimal $v$ have brought many questions about the so-called transport density, a measure of the local amount of traffic during the transportation which is naturally associated to these 1-homogeneous transport problems. I will present the problem using in particular some recent approach based on the flow by Dacorogna-Moser, and give the main results on the transport density.