In recent years, the Air Supported Vessels (ASVs) has received some interest due to increasing oil prices and the stricter regulations on emissions. The ASVs have the potential of reducing fuel consumption with less drag by adopting air cavity underneath its hull surface. Research so far has been mainly focused on the mechanism and effectiveness of the air cavity for drag reduction, i.e., the resistance in calm water condition. Other hydrodynamic performances of the ASVs are rarely studied. In this thesis, it is attempted to address some of the other hydrodynamic problems for the ASV, namely, wash wave field, motion response in waves and the stability. New mathematical models have been proposed to tackle these problems. The models cover the steady flow, frequency domain analysis (seakeeping) and time domain analysis (the dynamic stability). Emphases are placed on numerical calculation of the flow field generated by the excess pressure inside the cavity. Although the analytic expressions of the potential flow field by a pressure patch moving on the free surface are well known, the numerical calculations remain challenging. The singularities and highly oscillatory behaviour of the velocity potentials and the free surface elevations will cause numerical instability problem. In this study, new numerical schemes are proposed and the irregularities have been successfully removed. A number of case studies have been carried out to verify the proposed mathematical models and numerical methods. Satisfactory agreements have been found as far as there are other computations or measurements for comparison; or reasonable results are obtained. It is expected that the mathematical model, the numerical methods and tools established in the present study can be a supplementary means for developing the ASV at both design and operational stages.