This dissertation deals with some highly non-linear free surface flow problems. The first part is dedicated to the study of the bursting of a bubble at the free surface of a liquid. To this end, a numerical method is used which solves the Navier-Stokes equations, in the presence of a free surface. This free surface is described using a markers chain. The surface tension terms are then treated with high precision. A parametric study on the bubble radius is undertaken and a range of parameter region is found to correspond to a curvature singularity in finite time. The existence of this singularity is confirmed by a self-similar theory, based on an invicid formulation. The second part presents the resolution of potential axisymmetric flows in the presence of a free surface. The theory and developpment of a boundary integral method is described in details. Firstly, the method is validated using the theoretical knowledge of the spherical harmonics of an oscillating liquid drop. Then it is applied to two distinct problems : the impact of a drop on a super hydrophobic surface and the coalescence of two spherical drops. This second case presents a singularity in finite time because it involves a high curvature region at the time of contact. The scaling laws describing this singularity are then discussed and a comparison between experiments and numerical results is presented.
