A time domain mathematical model is presented for the prediction of ship motions in waves. This mathematical model incorporates convolution integrals, allowing fluid memory effects to be included in the modelling of ship responses to arbitrary excitation. The required impulse response functions are calculated from transforms of frequency domain data evaluated using a three dimensional potential flow analysis. The wave excitation impulse response functions include a negative time component, accounting for the influence of a particular wave before it reaches the reference point at midships. Impulse response functions are calculated with reference to both equilibrium and body fixed axis systems. Using reverse transforms, it is shown that certain frequency domain data, referenced to equilibrium axes, appears to be unsuitable for the calculation of impulse response functions. To test the effects of using impulse response functions referenced to either axis system, time domain simulations are performed using equations of motion referenced to both axis systems. Comparisons with frequency domain predictions show that the numerical implementation of the model referenced to body fixed axes is more accurate. Subsequently, non-linear incident wave and restoring force/moment contributions are included in the mathematical model referenced to body fixed axes. These contributions are accounted for by considering the instantaneous underwater portion of the hull at each time step of the simulations. Predictions for this partly non-linear model are compared to both linear predictions and experimental models for a range of wave amplitudes. A number of vessels types are considered, including vessels with flared hull forms and multihulls.