Underwater acoustic modelling is an important aspect of Synthetic Aperture Sonar (SAS) system design and algorithm development. Sea-trials are an expensive and time-consuming exercise and simulations provide an efficient and economic alternative. However, there are few simulators (in the open literature) that can efficiently provide realistic SAS data for large, complicated scenes. Conventional side-scan sonar simulators are not suitable for SAS data simulation. These simulators utilise narrow-beam and narrow-band approximations; typical SAS systems are wide-beam and wide-band and these approximations are invalid. Moreover, conventional side-scan sonar is a non-coherent imaging technique and SAS processing relies on the phase. Existing SAS simulators are capable of modelling very simple scenes only. They utilise a decomposition of the scene into point or smooth facet primitives, which is very inefficient. Many primitives are required and this imposes a severe restriction on scene complexity and size. This thesis presents a rigorous mathematical framework for the modelling of SAS imagery. A novel acoustic scattering model is developed and its implementation in a wide-beam and wide-band, multiple-receiver Interferometric SAS (InSAS) simulator is detailed. The scattering model utilises a decomposition of the scene into rough (rather than smooth) facet primitives. The use of rough facet primitives provides a significant increase in computational efficiency since scenes are decomposed into fewer primitives. This facilitates the simulation of larger and more complicated scenes. Each rough facet is characterised by its far-field beampattern. The statistics of the beampattern are related to the facet shape and roughness statistics using the Kirchhoff approximation. The beampattern is realised from its first and second-order statistics. The SAS imagery is obtained using a coherent sum of the facet responses and occlusions and multiple-scattering are resolved by ray-tracing. The simulator is implemented for use on a parallel computing cluster. The simulator is shown to provide realistic SAS data that is qualitatively and quantitatively similar to real data. The simulated results are considered, in many ways, superior to the simulated results in the literature.