We compare the ultrahigh vacuum growth of several different archetype organic molecular compounds on a variety of substrates, and find that the resulting thin films can form epitaxial, van der Waals epitaxial (vdWE) or quasi-epitaxial (QE) structures. Epitaxial and vdWE films form lattices commensurate with that of the substrate, where epitaxy refers to chemisorbed films while vdWE refers to purely vdW-bonded, physisorbed films. In contrast, QE films are incommensurate with the substrate lattice. In spite of this incommensurability, QE films can exhibit long range order, and have a unique orientational relationship with the substrate lattice. Quasi-epitaxial growth can result in strained structures, where the strain energy is relieved by variations in the internal lattice degrees of freedom. It has been found that QE growth can thus result in stable, quasi-equilibrium film structures which are different than the bulk structure of that same material. Organic thin films such as the phthalocyanines, 3,4,9,10-perylenetetracar☐ylic dianhydride, and the fullerenes grown on graphite, alkali halide crystals, Au, semiconductors and other substrates are all discussed.