In this talk, we describe and discuss elements of mathematical theory of the Hughes model for pedestrian evacuation. We present recent existence results obtained with M.D. Rosini, G. Stivaletta and with T. Girard for the one-dimensional case, either via convergence of the deterministic many-particle approximation or via the Schauder fixed-point approach, along with the limitations and the extensions of our approach. Some numerical simulations, in one and two directions, will also be presented and commented.