This book gives a survey of analytical, asymptotic, numerical and combined methods of solving eigenvalue problems. It considers a new method of accelerated convergence for solving problems of the Sturm-Liouville type as well as boundary-value problems with boundary conditions of the first, second and third kind. This volume also presents high-precision asymptotic methods for determining eigenvalues and eigenfunctions of higher oscillation modes and considers numerous eigenvalue problems that appear in oscillation theory, acoustics, elasticity, hydrodynamics, geophysics, quantum mechanics, structural mechanics, electrodynamics and microelectronics.