The essential spectrum of a Toeplitz operator generated by a continuous symbol is known to be equal to the range of the symbol. A much more difficult question is the characterization of the eigenvalues of Toeplitz operators. If we take a point in the spectrum that is not in the essential spectrum of the Toeplitz operator, then we can easily determine whether the point is an eigenvalue by using Coburn's Theorem. If the point is in the essential spectrum, then the question becomes much more delicate and only partial answers are known. I discuss the known results and compare the situation to that of the large truncated Toeplitz matrices.