We present fast, high-order integral equation methods for scattering by two- and three-dimensional inhomogeneous media. We first describe an extremely efficient, high-order accurate FFT-based method in three dimensions. The convergence rate of this approach depends on the regularity of the scatterer, achieving third-order convergence for discontinuous scatterers and superalgebraic convergence for smooth scatterers in the far field. This high-order accuracy is obtained through appropriate Fourier-smoothing of the discontinuous refractive index and through a partition of unity decomposition of the Green's function.

We next describe a method that combines the first approach with an exponentially accurate integrator for singular integrals to obtain an efficient, arbitrarily high-order accurate method. In more detail, we use a partition of unity decomposition of the scatterer, decomposing it into smooth bulk regions and discontinuous edge regions. The FFT-based method described above is used to integrate over the bulk regions and the exponentially accurate integrator is used to integrate over the edge regions. Finally, the contributions from the edge regions on the bulk regions is computed using an equivalent source acceleration strategy. We present several numerical examples including parallel computations to demonstrate the performance of these methods.