The transmission eigenvalue problem plays a central role in inverse scattering theory. This is a non-selfadjoint problem for a coupled pair of partial differential equations in a bounded domain corresponding to the support of the scattering object. Unfortunately, relatively little is known about the spectrum of this problem. In this talk I will consider the simplest case of the transmission eigenvalue problem for which the domain and eigenfunctions are spherically symmetric. In this case the transmission eigenvalue problem reduces to an eigenvalue problem for ordinary differential equations. Through the use of the theory of entire functions of a complex variable, I will show that there is a remarkable diversity in the behavior of the spectrum of this problem depending on the behavior of the refractive index near the boundary. Included in my talk will be results on the existence of complex eigenvalues, the inverse spectral problem and a remarkable connection (due to Fioralba Cakoni and Sagun Chanillo) between the location of transmission eigenvalues for automorphic solutions of the wave equation in the hyperbolic plane and the Riemann hypothesis.