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We report the growth of high-quality epitaxial MgB2 thin films on (1102) Al2O3 and (001) SrTiO3 substrates by using a pulsed laser deposition technique. The thin films grown on Al2O3 substrates show a Tc of 39 K with a sharp transition width of ~ 0.7 K. The critical current density in zero field is ~ 8 x 10^6 A/cm^2 at 5 K, suggesting that this compound has great potential for industrial applications. Also, Al2O3 is a very promising substrate in terms of superconducting device applications because of its high thermal conductivity and small dielectric constant. For the films deposited on Al2O3, X-ray diffraction patterns indicate a c-axis-oriented crystal structure perpendicular to the substrate surface whereas the films deposited on SrTiO3 show {101} plane as the preferred orientation.