  In this paper, we study the transmission eigenvalue problem for an
anisotropic material with a conductive boundary. We prove that the transmission
eigenvalues for this problem exist and are at most a discrete set. We also
study the dependence of the transmission eigenvalues on the physical parameters
and prove that the first transmission eigenvalue is monotonic. We then consider
the limiting behavior of the transmission eigenvalues as the conductive
boundary parameter $\eta$ vanishes or goes to infinity in magnitude. Finally,
we provide some numerical examples on three different domains to demonstrate
our theoretical results.
