Fluid-structure interaction (FSI), occurrent in many areas of engineering and in the natural world, has been the subject of much research using a wide range of modelling strategies. However, problems with high levels of structural deformation are difficult to resolve and this is particularly the case for biomedical flows. A Lagrangian flow model coupled with a robust model for nonlinear structural mechanics seems a natural candidate since large distortion of the computational geometry is expected. Smoothed particle Hydrodynamics (SPH) has been widely applied for nonlinear interface modelling and this approach is investigated here. Biomedical applications often involve thin flexible structures and a consistent approach for modelling the interaction of fluids with such structures is also required. The Lagrangian weakly compressible SPH method is investigated in its recent delta-SPH form utilising inter-particle density fluxes to improve stability. Particle shifting is also used to maintain particle distributions sufficiently close to uniform to enable stable computation. The use of artificial viscosity is avoided since it introduces unphysical dissipation. First, solid boundary conditions are studied using a channel flow test. Results show that when the particle distribution is allowed to evolve naturally instabilities are observed and deviations are noted from the expected order of accuracy. A parallel development in the SPH group at Manchester has considered SPH in Eulerian form (for different applications). The Eulerian form is applied to the channel flow test resulting in improved accuracy and stability due to the maintenance of a uniform particle distribution. A higher-order accurate boundary model is developed and applied for the Eulerian SPH tests and third-order convergence is achieved. The well documented case of flow past a thin plate is then considered. The immersed boundary method (IBM) is now a natural candidate for the solid boundary. Again, it quickly becomes apparent that the Lagrangian SPH form has limitations in terms of numerical noise arising from anisotropic particle distributions. This corrupts the predicted flow structures for moderate Reynolds numbers (O(102)). Eulerian weakly compressible SPH is applied to the problem with the IBM and is found to give accurate and convergent results without any numerical stability problems (given the time step limitation defined by the Courant condition). Modelling highly flexible structures using the discrete element model is investigated where granular structures are represented as bonded particles. A novel vector-based form (the V-Model) is identified as an attractive approach and developed further for application to solid structures. This is shown to give accurate results for quasi-static and dynamic structural deformation tests. The V-model is applied to the decay of structural vibration in a still fluid modelled using Eulerian SPH with no artificial stabilising techniques. Again, results are in good agreement with predictions of other numerical models. A more demanding case representative of pulsatile flow through a deep leg vein valve is also modelled using the same form of Eulerian SPH. The results are free of numerical noise and complex FSI features are captured such as vortex shedding and non-linear structural deflection. Reasonable agreement is achieved with direct in-vivo observations despite the simplified two-dimensional numerical geometry. A robust, accurate and convergent method has thus been developed, at present for laminar two-dimensional low Reynolds number flows but this may be generalised. In summary a novel robust and convergent FSI model has been established based on Eulerian SPH coupled to the V-Model for large structural deformation. While these developments are in two dimensions the method is readily extendible to three-dimensional, laminar and turbulent flows for a wide range of applications in engineering and the natural world.