The crystal structure and magnetic properties of epitaxial ZnO thin films doped with 5 at% Cu on SrTiO3 (001) and (111) substrates were investigated. In the,case of films deposited in oxygen, unique crystallographic growth directions on different substrates were observed, while a metallic phase was detected in films grown under vacuum. The Cu-doped ZnO thin films deposited on the SrTiO3 (111) substrates, with hexagonal structures, showed a single epitaxial relationship with the substrates, whereas those deposited on the SrTiO3 (001) substrates showed a double epitaxial growth mode. The epitaxial ZnCuO thin films deposited on the SrTiO3 (111) substrates under high vacuum exhibited a ferromagnetic signal at room temperature.