In this talk we will present some new algorithms that have been recently developed for the solution of electromagnetic and acoustic scattering problems and 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, especially at high frequencies, without sacrificing accuracy and error-controllability. In this connection, we shall first introduce a novel approach to the rigorous numerical solution of the integral-equation formulation of (surface) scattering problems in the high-frequency regime. As we will show, this 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 of configurations; examples from implementations of this approach in the context of bounded, unbounded (periodic) and multi-scale scattering surfaces will be presented. As we shall explain, these high-frequency integral-equation solvers possess the additional advantage that they seamlessly reduce to more standard schemes as the frequency decreases. Moreover, at the other end of the spectrum, the algorithms are based on the use of a geometrical optics (GO) ansatz (or of the related physical optics or geometrical theory of diffraction approximations) for the unknown surface currents and, thus, they naturally connect with classical approximate high-frequency solvers. With regards to the latter, time-permitting, we shall further review some recent developments on the numerical solution of the GO model itself, and we shall present a new (Eulerian) high-order accurate procedure for the solution of its phase-space formulation that is based on spectral and discontinuous Galerkin approximations.