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We present an overview of rational Krylov methods for solving the nonlinear eigenvalue problem (NLEP): A(λ)x = 0. For many years linearizations are used for solving polynomial eigenvalue problems. On the other hand, for the general nonlinear case, A(λ) can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. The major disadvantage of linearization based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, we introduce a new framework of Compact Rational Krylov (CORK) methods which maximally exploit the structure of the linearization pencils. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problems are negligible for large-scale problems.