  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.
