Wave breaking is a challenging subject that is not encompassed in the usual mathematical description of water waves. This is the consequence of the impossibility to represent the water-air interface as the graph of a function. In the first part of this presentation, we shall exhibit the strong non-linear nature of the breaking phenomena through the mathematical study of two water waves models: (1) KdV, whose solutions do not break and (2) Camassa-Holm, whose non-global solutions do break at some point. Next we shall discuss the way to incorporate multi-valued interfaces in the usual water waves problem before discussing whether or not this model efficiently describes the breaking phenomenon. In a third, ultimate, part we will present new numerical results that have been obtained using a finite-element code to solve the free-surface Navier-Stokes equations for an initial condition leading to a plunging jet. We compare these results with those obtained solving the Euler equation in the exact same configuration and conclude about the convergence of the two methods whenever the Reynolds number increases. We also discuss the influence of viscous dissipation on the overall shape of the wave.