Exact eigenvalues, eigenvectors, and principal vectors of operators with infinite dimensional ranges can rarely be found. Therefore, one must approximate such operators by finite rank operators, then solve the original eigenvalue problem approximately. This book addresses this issue of solving eigenvalue problems for operators on infinite dimensional spaces. From a review of classical spectral theory, through approximation techniques, to ideas for further research that would extend the results described, this volume serves as both a text for graduate students and as a source of state-of-the-art results for research scientists.