Variational inequalities provide the mathematical framework for a variety of nonlinear and non-smooth phenomena in science and engineering. Efficient and reliable solution procedures are closely linked to the mathematics of function spaces and partial differential equations. The main subjects of this book are a posteriori error analysis and convergence of adaptive finite element methods, the treatment of multigrid methods and multilevel methods with solution-dependent multilevel bases. The discussion is centered around some model obstacle and contact problems and based upon introductory sections on the mathematics of variational inequalities and on finite element discretisation including short MATLAB programs.