Catalysis by AuCu nanoclusters is a promising scientific field. However, our fundamental understanding of the underlying mechanisms of mixing in AuCu clusters at the sub-nanometre scale and their physico-chemical properties in both the gas-phase and on oxide supports is limited. We have identified the global minima of gas-phase and MgO(100)-supported AuCu clusters with 3–10 atoms using the Mexican Enhanced Genetic Algorithm coupled with density functional theory. Au and Cu adatoms and supported dimers have been also simulated at the same level of theory. The most stable composition, as calculated from mixing and binding energies, is obtained when the Cu proportion is close to 50%. The structures of the most stable free AuCu clusters exhibit Cu-core/Au-shell segregation. On the MgO surface however, there is a preference for Cu atoms to lie at the cluster-substrate interface. Due to the interplay between the number of interfacial Cu atoms and surface-induced cluster rearrangement, on the MgO surface 3D structures become more stable than 2D structures. The O-site of MgO surface is found to be the most favourable adsorption site for both metals. All dimers favour vertical (V) configurations on the surface and their adsorption energies are in the order: AuCu < CuCu < AuAu < AuCu (where the underlined atom is bound to the O-site). For both adatoms and AuCu dimers, adsorption via Cu is more favourable than Au-adsorbed configurations, but, this disagrees with the ordering for the pure dimers due to a combination of electron transfer and the metal-on-top effect. Binding energy (and second difference) and HOMO-LUMO gap calculations show that even-atom (even-electron) clusters are more stable than the neighbouring odd-atom (odd- electron) clusters, which is expected for closed- and open-shell systems. Supporting AuCu clusters on the MgO(100) surface decreases the charge transfer between Au and Cu atoms calculated in free clusters. The results of this study may serve as a foundation for designing better AuCu catalysts.