Nonlinear eigenvalue problems arise in many fields of natural and engineering sciences. Theoretical and practical results are scattered in the literature and in most cases they have been developed for a certain type of problem. In this book we consider the most general nonlinear eigenvalue problem without assumptions on the structure or spectrum and provide basic facts. Nonlinear Rayleigh functionals are analyzed in detail and in different forms. New methods for the computation of eigenvalues and -vectors are designed based on Rayleigh functionals, and well-known Jacobi--Davidson methods are discussed. Asymptotically cubic convergence of the two-sided nonlinear Jacobi--Davidson method is shown. The special case of nonlinear complex symmetric eigenvalue problems is examined. We show the appropriate definition of a complex symmetric Rayleigh functional, which is used to derive a complex symmetric Rayleigh functional iteration which converges locally cubically, and the complex symmetric residual inverse iteration method. All methods are also illustrated numerically. The book is addressed to mathematicians as well as engineers interested in nonlinear eigenvalue problems.