In this study results are reported on the modelling of the wave drift forces on a vessel in regular deep water waves with forward speed. In chapters 3 and 5 the mathematical background of the boundary integral method for the computation of the first order and second order wave drift forces on floating bodies at low forward speeds is presented. By means of the Green's theorem a source distribution is derived. The Green's function (source function) and source strength are evaluated asymptotically for small values of the forward velocity. Also, the forward speed Green's function is linearised with respect to forward speed. The first two terms of the source strength over the mean wetted surface of the body is then computed from two sets of integral equations. The kernel of these sets of integral equations has the same form as the integral equation for the zero speed problem. In addition to the zero speed problem, a free surface integral enters the right hand side of the integral equation for the source strengths. In the development of the linearised forward speed Green's function with respect to forward speed, corrections on the asymptotic approximation are also given in order to arrive at a proper uniform expansion with respect to forward speed. Since the encounter frequencies are usually higher at forward speed than at zero speed for head on and bow quartering waves, the effect of 'irregular frequencies' is also described. A robust lid method is put forward to solve the effects of this 'irregular frequency' problem. The mean wave drift forces are found by a far field analysis. The results of the wave drift forces on a floating sphere in regular waves compare favourably to the results of the study of Zhao and Faltinsen. To validate the approach for the determination of the wave drift forces, model test experiments were performed on a 200 kDWT tanker in fully loaded as well as ballast condition. From the comparison with the results of model tests it is concluded that the linearised forward speed description works well for head current cases and the tested wave directions. Less good agreement is found when the current is coming from a bow quartering direction.