The effect of damage induced by 2-MeV alpha particles, followed by annealing, on the critical temperature (Tc), resistivity, and upper critical field (Hc2), of three MgB2 films made by different deposition processes has been studied. Damage creates a linear decrease in Tc with residual resistivity, and produces maxima in both Hc2(0)-perpendicular and Hc2(0)-parallel. Below Tcs of about 25 K, Hc2(0) depends roughly linearly on Tc, while the anisotropy of Hc2(0) decreases as Tc decreases. Annealing the films reproduces the Tc vs. residual resistivity dependence but not the Hc2(0) values induced by damage.