We studied structure and reactivity of ZnO(0001) ultrathin films grown on Ag(111) and Cu(111) single crystal surfaces. Structural characterization was carried out by scanning tunneling microscopy, Auger electron spectroscopy, low-energy electron diffraction, and temperature programmed desorption. The CO oxidation behavior of the films was studied at low temperature (450 K) at near atmospheric pressures using gas chromatography. For ZnO/Cu(111), it is shown that under reaction conditions ZnO readily migrates into the Cu crystal bulk, and the reactivity is governed by a CuOx oxide film formed in the reaction ambient. In contrast, the planar structure of ZnO films on Ag(111) is maintained, similarly to the previously studied ZnO films on Pt(111). At sub-monolayer coverages, the "inverse'' model catalysts are represented by two-monolayer-thick ZnO(0001) islands on Pt(111) and Ag(111) supports. While the CO oxidation rate is considerably increased on ZnO/Pt(111), which is attributed to active sites at the metal/oxide boundary, sub-monolayer ZnO films on Ag(111) did not show such an effect, and the reactivity was inhibited with increasing film coverage. The results are explained by much stronger adsorption of CO on Pt(111) as compared with Ag(111) in proximity to O species at the oxide/metal boundary. In addition, the water-gas shift and reverse water-gas shift reactions were examined on ZnO/Ag(111), which revealed no promotional effect of ZnO on the reactivity of Ag under the conditions studied. The latter finding suggests that wetting phenomena of ZnO on metals does not play a crucial role in the catalytic performance of ZnO-based real catalysts in those reactions.