This thesis considers a foil propulsion system on a supply vessel. In analysing the potential of a foil propulsion system, it is imperative to establish a rigid mathematical model. In that respect, modelling of the dynamic system is emphasised, and a comprehensive study is presented on the matter. The equations of motion for an oscillating foil and a vessel are derived, separately. The two systems are then combined, to form the coupled vessel-foil structure. For the vessel, a time-domain model based on Cummins' equation is proposed. Cummins' equation has proven efficient in assessing a unified seakeeping and manoeuvring problem (Fossen [2011]). In line, the vessel-foil system will be exposed to both vessel oscillatory motion due to waves and forward speed effects, i.e seakeeping and manoeuvring. Moreover, the efficiency of the foil is directly dependent on the two.Additionally, aspects of foil control is looked into. In theory, active control could maximise thrust while preventing stall, which would be ideal. The validity of simulations with active control, however, is highly dependent on the accuracy of the emph{basic} vessel-foil model. Consequently, effort has been focused on presenting a rigid mathematical foundation.