In this course, we present the state of the art for linear as well as nonlinear multilevel and multigrid methods. The solution of large linear and nonlinear systems of equations is one of the most important tasks in numerical simulation. Since standard solution methods do not scale optimally , alternative solution strategies have been developed during the last decades. In particular multilevel or multiscale solution strategies have been developed, which are often employed due to their high efficiency. Prominent examples are multilevel or domain decomposition methods for linear elliptic problems, In this course, we start from well known subspace correction methods for linear problems and proceed to more recent developments as are nonlinear multigrid and monotone multigrid. Finally, we will consider (recursive) trust-region methods and their application to minimization problems in computational mechanics. For all methods, we will also discuss their parallelization.