This article reviews the solution procedures used to extract eigenvalues and eigenvectors from linear dynamic systems using the finite element method. The main focus of this review article is on eigensolution techniques that provide only a partial eigensolution. Such eigensolvers extract only a small subset (normally the lowest) of the eigenvalues and eigenvectors present in the discretized system. They represent the most efficient approach to extracting eigenvalues and eigenvectors from large degree of freedom systems. The techniques covered include the subspace iteration method, the Lanczos method, the conjugate gradient method, the Ritz vector method, the substructure synthesis methods (including component mode synthesis), and the condensation techniques (including dynamic substructuring). Each subsection contains a variety of references that cover the current state of research as well as the origin of each technique. A brief critique of the eigensolution procedures used for solving small (less than 250 degree of freedom) eigenproblems is also included for completeness. The discussion is generally limited to eigensolution techniques for linear undamped eigenproblems of the form AΦ = CΦΛ, where A and C are frequently symmetric and positive definite. A brief review of eigensolvers for the nonsymmetric eigenproblem, such as that arising in damped structures, is included at the end. This review article contains 320 references.