We examine the synchronization phenomena in a set of globally coupled oscillators with particular attention to the interplay between external driving and inertia or noise. From the self-consistency equation for the order parameter, which measures the collective synchronization of the system, it is found that the inertia brings on discontinuous transitions between the coherent and incoherent states and gives rise to hysteresis behavior. It is also shown that the total order parameter decreases monotonically with noise, indicating overall suppression of synchronization. Still there exist optimal noise levels at which the ac components of the order parameter and of the phase velocity reaches their maxima. ©2000 American Institute of Physics.