In this talk, we will investigate the acoustic transmission eigenvalue problem associated with an inhomogeneous media with a conductive boundary. These are a new class of eigenvalue problems that are not elliptic, not self-adjoint, and nonlinear, which gives the possibility of complex eigenvalues. The talk will consider the case of an Isotropic and Anisotropic scatterer. We will discuss the existence of the eigenvalues as well as their dependence on the material parameters. Because this is a non-standard eigenvalue problem, a discussion of the numerical calculations will also be highlighted. Lastly, we will discuss recovering the scatterer using a monotonicity method that is independent of the transmission eigenvalues. This is joint work with: O. Bondarenko, V. Hughes, A. Kleefeld, H. Lee, and J. Sun.