Two members of the copper-containing amine oxidase family are physiologically important proteins: (1) Diamine oxidase (hDAO; AOC1) with a preference for diamines is involved in degradation of histamine and (2) Vascular adhesion protein-1 (hVAP-1; AOC3) with a preference for monoamines is a multifunctional cell-surface receptor and an enzyme. hVAP-1-targeted inhibitors are designed to treat inflammatory diseases and cancer, whereas the off-target binding of the designed inhibitors to hDAO might result in adverse drug reactions. The X-ray structures for both human enzymes are solved and provide the basis for computer-aided inhibitor design, which has been reported by several research groups. Although the putative off-target effect of hDAO is less studied, computational methods could be easily utilized to avoid the binding of VAP-1-targeted inhibitors to hDAO. The choice of the model organism for preclinical testing of hVAP-1 inhibitors is not either trivial due to species-specific binding properties of designed inhibitors and different repertoire of copper-containing amine oxidase family members in mammalian species. Thus, the facts that should be considered in hVAP-1-targeted inhibitor design are discussed in light of the applied structural bioinformatics and structural biology approaches.