An important predicted, but so far uncharacterized, property of the new superconductor MgB2 is electronic anisotropy arising from its layered crystal structure. Here we report on three c-axis oriented thin films, showing that the upper critical field anisotropy ratio Hc2par/Hc2perp is 1.8 to 2.0, the ratio increasing with higher resistivity. Measurements of the magnetic field-temperature phase diagram show that flux pinning disappears at H* ~ 0.8Hc2perp(T) in untextured samples. Hc2par(0) is strongly enhanced by alloying to 39 T for the highest resistivity film, more than twice that seen in bulk samples.