We first derive the equations for weak noise perturbations of exponentially stable limit cycles. With this perturbation theory, it become possible to compute quantities such as Liapunov exponents, di usion constants, and the effects of noise on frequency. We show that there are resonances between the frequency of the oscillations and the time scale of the noise. Next we apply this theory to synchronization of oscillators that receive partially correlated noise. We derive the invariant density and order parameters for the degree of synchronization. We show that some types of oscillators are better synchronizers than others. We conclude with some surprising eff ects of heterogeneity on the synchronization.