As a result of studies of the far field pattern of the
scattered wave for time harmonic acoustic and electromagnetics, a
new class of interior problem arises termed the "Interior
Transmission Problem" (ITP). The ITP is not a standard elliptic
problem, and a study of the solvability of this problem gives rise
to a non-standard eigenvalue problem for the ITP. The proof of
existence and properties of these eigenvalues is not
straightforward. I shall survey the ITP and its properties in
several applications, and describe numerical schemes for computing
transmission eigenvalues. Remarkably, transmission eigenvalues can
be observed from far field data, and the resulting eigenvalues can
be used to estimate properties of the scatterer. The interior
transmission problem has similarities with problems involving
negative index of refraction, so it may be that the study of
interior transmission eigenvalues is also relevant there.
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