The interior transmission problem plays a basic role in the investigation of inverse scattering problems for inhomogeneous medium. In the paper the interior transmission problem for the Maxwell equations in the electromagnetic scattering problem for an anisotropic inhomogeneous object is studied. The authors explain how the interior transmission problem arises when one considers the corresponding inverse scattering problem. A variational approach is developed to investigate the case when the index of refraction is less then one as well as for partially coated scatters. The authors show how the variational approach leads to Fredholm equations. The structure of the transmission eigenvalues is described.