For acoustic scattering we consider the inverse problem for a complex scatterer, which might be composed of both inhomogeneous anisotropic mediums and impenetrable obstacles. It is shown that the fixed-frequency scattering amplitude uniquely determines the scattering support of a complex scatterer, disregarding its contents. We also study a special kind of transmission problem.

For electromagnetic scattering, we develop a direct scheme of imaging multiple electromagnetic scatterers. The multiple scatterers may consists of both inhomogeneities and impenetrable perfect electromagnetic conductive obstacles. Moreover, there might be extended components presented simultaneously in the multiple scatterers. The proposed reconstruction scheme is based on an imaging functional which can be directly calculated without any inversion involved. We also provide a heuristic principle study for this imaging scheme. We also provide extensive numerical study to illustrate the effectiveness and efficiency of the scheme.