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Co epitaxial thin films were prepared on single-crystal MgO substrates of (100), (110), and (111) planes heated at 300degC by ultra high vacuum molecular beam epitaxy. The film crystal structure varies depending on the substrate orientation. On MgO(100) substrate, a Co(112macr0) epitaxial bi-crystalline film with an hcp structure is obtained. A Co(110) single-crystal film with an fcc structure is formed on MgO(110) substrate, whereas a Co(0001) single-crystal film with an hcp structure is grown on MgO(111) substrate. Co film growth on MgO substrates follows the Volmer-Weber mode. The in-plane magnetization properties reflect the magnetocrystalline anisotropy of bulk hcp-Co or fcc-Co crystal. The Gilbert's damping constants of these 40-nm-thick epitaxial Co films are determined.