We have studied the adsorption of associating fluids on solid substrates, in the framework of the models with two and four bonding sites and with nonassociative Lennard-Jones (LJ) attraction between fluid particles. The fluid-substrate interaction is chosen in the form of the 9-3 LJ-type potential. Our analysis involves the strongly adsorbing, graphitelike substrate, as well as several weaker model surfaces. Special attention is paid to the layering transitions, their relevance to the prewetting transition, and the prewetting transition on its own. We have evaluated the surface critical temperature and made estimates for the critical temperatures of the layering transitions and of the wetting temperature, for each model in question. An insight into thermodynamic behavior of the models is given in terms of the phase diagrams. Our calculations reveal that the details of the surface phase behavior of strongly associating fluids differ from the phase behavior for weakly associating fluids and of; adsorbed, simple nonassociating fluid. Also, we have analyzed structural changes in the adsorbed films during transitions in question, considering the entire distribution of fluid particles and a set of distributions for fractions of differently bonded species.