We fabricated high-quality c axis-oriented epitaxial MgB2 thin films using a pulsed laser deposition technique. The thin films grown on (1 i 0 2) Al2O3 substrates have a transition temperature of 39 kelvin. The critical current density in zero field is approximately 6 x 10(6) amperes per cubic centimeter at 5 kelvin and approximately 3 x 10(5) amperes per cubic centimeter at 35 kelvin, which suggests that this compound has potential for electronic device applications, such as microwave devices and superconducting quantum interference devices. For the films deposited on Al2O3, x-ray diffraction patterns indicate a highly c axis-oriented crystal structure perpendicular to the substrate surface.