  In this paper, we consider the inverse scattering problem associated with an
inhomogeneous media with a conductive boundary. First, we discuss the inverse
conductivity problem of reconstructing the conductivity parameter from
scattering data. Next, we consider the corresponding interior transmission
eigenvalue problem. This is a new class of eigenvalue problem that is not
elliptic, not self-adjoint, and non-linear, which gives the possibility of
complex eigenvalues. We investigate the convergence of the eigenvalues as the
conductivity parameter tends to zero as well as prove existence and
discreteness for the case of an absorbing media. Lastly, several numerical and
analytical results support the theory and we show that the inside-outside
duality method can be used to reconstruct the interior conductive eigenvalues.
