I will discuss certain systems of transport type whose coefficients depend nonlinearly on the solution. Applications of such systems range from the modeling of pressure-less gases to the study of mean field games in a discrete state space. I will identify a notion of weak solution within the class of coordinate-wise decreasing functions, a condition which has particular relevance for applications arising in economics. I demonstrate the existence of a unique maximal and minimal solution, in an appropriate sense, and the discrepancy between the two can be precisely related to the formation of shocks. I will also present a selection principle for the family of solutions based on vanishing (nonlinear) viscosity. The analysis depends on new well-posedness results for linear transport equations with certain rough velocity fields, which are of independent interest. This is joint work with P.-L. Lions.