The scattering and trapping of oblique surface gravity waves by two classes of submerged topography is considered in this paper. Attention initially focuses on ridges of arbitrary, but uniform, cross-section and finite extent which protude from an otherwise flat bed. A more general shoaling topography is also considered in which an arbitrary bed profiles connects two semi-infinite regions of constant depth. In both cases, the aim has been to develop a new approach to solving these problems based on integral equation techniques. These are designed to be exact, in the context of the linear theory being used, and to generate integral equations which are only weakly singular by use of a novel transformation which converts normal derivatives to tangential derivatives. Two methods of numerical solution are implemented. A boundary element approach provides a remarkably effective method which can typically achieve three significant figure accuracy by inverting a 100x100 system of equations whose elements are easy and quick to calculate. In contrast, a Rayleigh-Ritz approach typically achieves higher accuracy by inverting a much smaller 10x10 real symmetric system of equations whose elements require more work to calculate. The methods are applied to a range of examples.