  We consider the scattering of waves by a penetrable inclusion embedded in
some reference medium. We exhibit examples of materials and geometries for
which non-scattering frequencies exist, i.e., for which at some frequencies
there are incident fields which produce null scattered fields outside of the
inhomogeneity. We show in particular that certain domains with corners or even
cusps can support non-scattering frequencies. We relate the latter, for some
inclusions, to resonance frequencies for Dirichlet or Neumann cavities. We also
find situations where incident non-scattering fields solve the Helmholtz
equation in a neighborhood of the inhomogeneity and not in the whole space.
Finally, in relation with invisibility, we give examples of inclusions of
anisotropic materials which are non-scattering for all real frequencies. We
prove that corresponding material indices must have a special structure on the
boundary.
