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 struct.ure or spectrum. We start by providing basic facts on the conditioning of a simple eigenvalue and an inverse operator representation in terms of the singular value decomposition. The main part of this work connects Newton-type methods for nonlinear eigenvalue problems and nonlinear Rayleigh functionals.