We discuss the state of the art in numerical solution methods for large scale polynomial or
rational eigenvalue problems. We present the currently available solution methods such as
the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties.
We briefly introduce a new linearization technique and demonstrate how it can be used to
improve structure preservation and with this the accuracy and efficiency of linearization based
methods. We present several recent applications where structured and unstructured nonlinear
eigenvalue problems arise and some numerical results.