In this talk we will present a variety of techniques for the solution of electromagnetic and acoustic scattering problems that are aimed at overcoming the limitations of state-of-the-art scattering solvers. We will begin with a brief review of the techniques most commonly used for the numerical simulation of scattering experiments, highlighting their advantages and shortcomings. In addition to providing a context for the presentation, the review will motivate the continued need for algorithms that can tackle these problems efficiently without sacrificing accuracy and error controllability. In this connection, we shall first discuss some theoretical considerations that lead to a class of effective, high-order perturbative methods that are particularly well-suited for rough-surface scattering (e.g. ocean surfaces, diffraction gratings, etc). We shall further explain how these algorithms can be used to resolve the scattering off multi scale surfaces, leading to consideration of high-frequency problems. With regard to the latter, we shall next present the main ideas behind our recent design of a technique for the solution of the integral-equation formulation of the scattering problem in the high-frequency regime (see also C. Turc's presentation for an application of these ideas in the context of three-dimensional rough-surface scattering). As we will show, our scheme can deliver error controllable answers without the need to discretize on the scale of the wavelength of radiation, and it therefore holds significant promise for applicability in a variety configurations (such as military vehicles illuminated with, e.g., centimeter waves). Finally, as we shall explain, our high-frequency strategy possesses the additional property that it continuously reduces to a full discretization of the integral equations as the frequency is decreased to a moderate regime. Our work on the design of effective high-order solvers for volumetric scattering applications in this range of moderate frequencies will be reviewed in M. Hyde's presentation.