This book provides an analysis of the boundary element method for the numerical solution of Laplacian eigenvalue problems. The representation of Laplacian eigenvalue problems in the form of boundary integral equations leads to nonlinear eigenvalue problems for related boundary integral operators. The concept of holomorphic Fredholm operator functions is used for the analysis of the boundary integral formulations of Laplacian eigenvalue problems. A convergence and error analysis for the Galerkin approximation of eigenvalue problems for holomorphic coercive operator functions is established. These results are applied to the Galerkin boundary element discretization of Laplacian eigenvalue problems. Different methods for the solution of algebraic nonlinear eigenvalue problems such as inverse iteration, Rayleigh functional iterations and Kummer's method are presented. For the latter method a numerical analysis for simple and multiple eigenvalues is given. In a numerical example, a boundary element and a finite element approximation of a Laplacian eigenvalue problem are compared. The theoretical results of the analysis of the boundary element method could be confirmed.