The focus of the research described herein is the scattering of time-harmonic electromagnetic waves when encountering with impenetrable and penetrable obstacles. We study both the direct and inverse problems. In the case of an impenetrable obstacle, we assume perfectly conducting boundary condition and apply the integral equation method to show well-posedness of the direct problem. In the case of a penetrable obstacle, we assume conducting transmission conditions and apply both the integral equation and variational method to show well-posedness. The inverse problem we consider is determining the shape of an obstacle from the knowledge of the far field pattern. Specifically, we concentrated on uniqueness issues, that is, we examined under what conditions an obstacle can be identified from a knowledge of its far far field patterns for incident plane waves. We conclude this thesis with a discussion of an interior eigenvalue problem motivated by the penetrable case with conducting boundary conditions and show that the set of transmission eigenvalues form at most a discrete set
