1 Introduction.- 2 Finite element approximation.- 2.1 Finite elements, multivariate splines, wavelets.- 2.2 Moduli of smoothness and K-functionals.- 2.3 Jackson and Whitney inequalities.- 2.4 Bernstein inequalities and inverse estimates.- 2.5 Information on other approximation schemes.- 2.6 Constructive characterization of Besov spaces.- 3 Function spaces.- 3.1 Spaces on Rd.- 3.2 Spaces on domains and extension.- 3.3 Spaces on manifolds and traces.- 3.4 Approximation spaces on polyhedral domains.- 4 Applications to multilevel methods.- 4.1 The abstract Schwarz theory.- 4.2 Second-order elliptic equations.- 4.3 The biharmonic problem.- 4.4 Domain decomposition and boundary element methods.- 4.5 Sparse grids.- 4.6 Nonconforming and mixed methods.- 5 Error estimates and adaptivity.- 5.1 Traditional error estimates.- 5.2 h-version and nonlinear approximation.- 5.3 Adaptive multilevel methods.- 5.4 More complicated approximation schemes.- References.