MgB2 thin films were cold grown on sapphire substrates by pulsed laser deposition followed by postannealing in mixed, reducing gas, Mg rich, Zr gettered, environments (pO(2)similar to 10(-24) atm) at 750 and 950 degreesC. The films had T-c in the range 29-34 K, J(c) (20 K, H=0) in the range 3x10(4)-3x10(5) A cm(-2), and irreversibility fields H-* at 20 K of 4-6.2 T. An inverse correlation was found between T-c and H-*. The films had grain sizes of similar to0.1-1 mum and a strong biaxial alignment was observed in the 950 degreesC annealed film. (111) oriented MgO was also observed. Mg coating of films during crystallization appeared to improve film T-c. (C) 2001 American Institute of Physics.