The diffraction of both scalar and vector monochromatic waves by totally-reflecting bodies is considered from a computational viewpoint. Both direct and inverse scattering are covered. By invoking the optical extinction theorem (extended boundary condition) the conventional singular integral equation (for the density of reradiating sources existing in the surface of the scattering body) is transformed into infinite sets of non-singular integral equations - called the null field equations. There is a set corresponding to each separable coordinate system. Each set can be used to compute the scattering from bodies of arbitrary shape but each is most appropriate for particular types of body shape, as is confirmed by computational results. The general null field is extended to apply to multiple scattering bodies. This permits use of multipole expansions in a computationally convenient manner, for arbitrary numbers of separated, interacting bodies of arbitrary shape. The method is numerically investigated for pairs of elliptical and square cylinders. A generalisation of the Kirchoff, or physical optics, approach to diffraction theory is developed from the general null field method. Corresponding to each particular null field method is a physical optics approximation, which becomes exact when one of the coordinates being used is constant over the surface of the scattering body. Numerical results are presented showing the importance of choosing the physical optics approximation most appropriate for the scattering body concerned. Generalised physical optics is used to develop two inversion procedures to solve the inverse scattering problem for totally-reflecting bodies. One is similar to conventional methods based on planar physical optics and, like them, requires scattering data at all frequencies. The other enables shapes of certain bodies of revolution and cylindrical bodies to be reconstructed from scattered fields observed at two closely spaced frequencies. Computational results which confirm the potential usefulness of the latter method are presented.