This talk addresses the mathematical analysis of fluid models including a maximum packing constraint. These equations arise naturally for instance in the modeling of mixtures like suspensions or in the modeling of collective motion. I will present recent existence and stability results on two classes of PDEs systems corresponding to two modeling approaches. The first one, called the "soft approach", is based on compressible equations complemented with constitutive laws that are singular close to the maximal packing constraint. The second one, the "hard approach", is based on a free boundary problem between a congested domain with incompressible dynamics and a free domain with compressible dynamics.