Ni and Cu thin films of about 100 and 50 nm were deposited on MgO(001), (110), (111), and diamond(001) substrates by using a molecular beam epitaxy (MBE) method. Their growth mechanisms were investigated by in-situ RHEED. We found that Ni films can be grown easily on a diamond(001) substrate where the lattice mismatch between Ni and diamond is very small (1.42%). We found that, despite a large lattice mismatch of 19.5% (16.5%) between Ni (Cu) and MgO, Ni and Cu films can be grown epitaxially on MgO(001), (110), and (111) substrates. Moreover we measured magneto-optical (MO) Kerr ellipticity spectra on the epitaxial Ni films on MgO and diamond substrates. As a result, no difference of MO Kerr spectra were observed on Ni(001), (110), and (111). The epitaxial Ni films with Cu buffer layers show relatively small coercive force (Hc) compared with that of epitaxial Ni films without Cu buffer layers.