This book consists of a collection of carefully written lecture notes delivered as part of the special semester on computational mechanics held in 2005 at the Radon Institute for Computational and Applied Mathematics. It provides valuable overviews of recent developments in four topics. The first lecture (by B. Kaltenbacher and M. Kaltenbacher, on modelling and iterative identification of hystersis via Preisach operators in partial differential equations) deals with mathematical modelling of hysteresis effects, occurring in many different areas in mechanics. In the second lecture (entitled ‘Multilevel methods for anisotropic elliptic problems’), J. Kraus and S. Margenov present various preconditioners based on optimal complexity multilevel methods suitable for problems with anisotropy in the mesh and in coefficients of a discrete problem. The third lecture (by S. Nepomnyaschikh on ‘Domain decomposition methods’) presents methods for developing efficient solvers for large scale numerical problems on massively parallel computers using the domain decomposition method and the fictious space method. The last lecture (by S. Repin with the title ‘A posteriori error estimation methods for partial differential equations’) gives a nice overview of methods for deriving a posteriori error estimates for partial differential equations especially for FEM discretizations. The first part of this lecture summarizes standard approaches to deriving a posteriori error estimates. In the second part, a recent method (free of any mesh-dependent parameters) is presented.