This thesis describes a finite element method for simulation of free-surface flows, such as ocean waves, using the discontinuous Galerkin method. Free-surface flows where there is a large difference in density between two immiscible fluids will have a large jump in the magnitude of the momentum field at the interface between the two fluids. Such discontinuities create problems for higher-order discretisations, which are more computationally efficient where the solution is smooth, but need careful handling near discontinuities in order to avoid Gibbs oscillations. This thesis shows how slope limiting can be used to stabilise the momentum equation in an exactly mass conserving discontinuous Galerkin finite element discretisation of the Navier–Stokes equation. The stabilised method is tested on a range of well known 2D and 3D free-surface-flow test cases. The results are in good agreement with published experimental results, and also give the expected higher-order convergence rates for smooth solutions.