In this talk we consider a class of scalar nonlinear models describing crowd dynamics. The congestion term appears in the transport equation in the form of a compactly supported nonlinear mobility function, thus making standard weak-type compactness arguments and uniqueness of weak solutions fail. We introduce two different approaches to the problem and discuss their connections with the well-posedness of entropy solutions of the target PDE in the sense of Kružkov. A deterministic particle approach relying on suitable generalisations of the "follow-the-leader" scheme, which can be interpreted as the Lagrangian discretisations of the problem; and a variational approach in the spirit of a minimising movement scheme exploiting the gradient flow structure of the evolution in a suitable metric framework.