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MgB2 thin film growth on sapphire (0001) and MgO substrates is reported. The thin films were deposited by using the co-evaporation method, in which the deposition rates were well controlled separately. The as-grown thin films demonstrated superconductivity without the use of any post-annealing process. The critical temperature dependence of the substrate temperature and the evaporation rates were investigated, and it was found that a high substrate temperature and high deposition rates are needed to produce high-quality films. Below a substrate temperature of 250°C, the films exhibited no x-ray diffraction peaks, but above it, the films tended to grow epitaxially to c-axis on sapphire (0001) substrate. The critical temperature of MgB2 film was over 30 K, and MgB2 thin films made by co-evaporation method are expected to have excellent properties after further optimization.