In this article we review the complexation behavior of charged polymers adsorbed onto charged substrates. In the first part we summarize the results obtained on the adsorption properties of a single polyelectrolyte chain with one or several oppositely charged spherical colloids. The results of various simulational studies are compared with theoretical predictions. After that, we show that under strong electrostatic coupling conditions one can also obtain complexations, where chain and colloid carry the same charge (like-charge complexation). This effect is due to counterion correlation effects. Finally, we investigate the case of polyelectrolyte multilayers structures, which are composed of alternating polycations and polyanions on various substrate geometries. This structure, normally obtained by a so-called layer-by-lay deposition, is very stable experimentally, but only poorly understood in terms of theoretical approaches. Our simulation results suggest that short range attractive interactions (e.g., van der Waals) are important for obtaining a stable sequence of multilayers. (C) 2004 Wiley Periodicals, Inc.