Theoretical descriptions of static properties of polymer brushes are reviewed, with an emphasis on monodisperse macromolecules grafted to planar, cylindrical, or spherical substrates. Blob concepts and resulting scaling relations are outlined, and various versions of the self-consistent field theory are summarized: the classical approximation and the strong stretching limit, as well as the lattice formulation. The physical justification of various inherent assumptions is discussed, and computer simulation results addressing the test of the validity of these approximations are reviewed. Also, alternative theories, such as the single chain mean field theory and the density functional theory, are briefly mentioned, and the main facts about the models used in the computer simulations are summarized. Both molecular dynamics and Monte Carlo simulations are described, the latter including lattice models and bead-spring models in the continuum. Also extensions such as brushbrush interactions or nanoparticles inside of brushes as well as the solubility of free chains in brushes are briefly mentioned. Pertinent experimental results, though still somewhat scarce, are mentioned throughout and their consequences on the status of the theoretical understanding of polymer brushes is emphasized. (C) 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012