Matrices and their associated linear transformations pop up frequently in signal processing and information theory. One such matrix is the Toeplitz matrix -- a matrix for which each descending diagonal is constant. Its associated linear transformation is the Toeplitz operator. The eigenvalues and spectrum for these operators have been a source of intrigue for a long while. Arising via specialized interpolation problems is a subclass of these operators called constrained Toeplitz operators. In this talk we introduce these Toeplitz operators, look at their eigenvalues, and attempt to draw topological conclusions about them. This discussion is based on joint work with C. Felder and B. Russo.