A nonlinear matrix is a matrix whose entries are nonlinear functions of a parameter. Such problems arise from physics (Schroedinger equation) and mechanical engineering (porous materials, boundary element method, e.g.). The last 20 years, rational approximation methods and linearization were proposed to approximate such matrices by linear pencils of much higher dimensions. Applications are the nonlinear eigenvalue problem, parametric linear systems, frequency sweeping, model order reduction and the solution of time dependent problems with nonlinear frequency dependencies. We give an overview of approximation methods with focus on AAA and Krylov methods that exploit the structure of the linear pencil.