A series of ab initio simulations, based on density functional theory, of the structure of the clean GaAs-(001)-(2 x 4) surface and of C2H2, C2H4, and trimethylgallium (TMGa) adsorbates are described. This surface was selected because of its importance in the growth of GaAs by molecular beam epitaxy. After summarizing briefly the theoretical basis of the computational methods used in the paper, we review critically what is known from experiment and theory about the structure of the clean surface. We argue that there is now strong evidence in favor of the "trench dimer" model for the beta-phase of the clean surface, while the structures of the alpha and gamma phases are less settled. We then present nb initio simulations of the trench dimer, the three dimer, and the gallium rebonded models of the clean GaAs(001)-(2 x 4) surface and discuss their common structural and bonding features. Ab initio simulations of C2H2 and C2H4 adsorbates at arsenic dimers of the GaAs(001)-(2 x 4) surface are then presented. The changes in the bonding configurations of both the adsorbates and the surface arsenic dimers are explained in terms of changes in the bond orders and local hybridization states.