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We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell's equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero.

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in R 2 and R 3 . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

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