Ordered monolayer and multilayer thin films of perylene-3,4:9, 10-tetracarboxylic dianhydride (PTCDA) have been grown on Cu(100) and MoS2(0001) single crystals. Low-energy electron diffraction (LEED) investigations have shown that PTCDA grows on Cu(100) in a commensurate rectangular lattice (4 root 2x5 root 2) = 45 degrees, whose dimensions (b(1) = 14.5 Angstrom, b(2) = 18.1 Angstrom) are significant different from those seen on the (0001) cleavage faces of single-crystal MoS2, where a coincident rectangular lattice with b(1) = 13.1 Angstrom and b(2) = 21.2 Angstrom is formed. This latter structure is much closer to the packing in the (102) plane of the bulk crystal of PTCDA. X-ray photoelectron spectra (XPS) and thermal desorption mass spectroscopy (TDMS) show that the first monolayer of PTCDA is strongly chemisorbed on the Cu surface. During the reaction with the copper surface each PTCDA molecule apparently loses the two bridging oxygen atoms in the anhydride groups, leading to a molecular system with close to the same dimensions and symmetry as the parent molecule. XPS data show that the adsorption on the surface of the weakly interacting metal dichalcogenide MoS2 is not accompanied by such a surface reaction. Support for this hypothesis was obtained by deposition of N,N’-dimethylperylene-3,4:9,10-bis(dicarboximide) (Me-PTCDI) on the Cu(100) surface. Me-PTCDI forms a rectangular (6 x 8) structure on Cu(100), with b(1) = 15.3 Angstrom and b(2) = 20.4 Angstrom.