Motivated by recent experiments on persistent currents in mesoscopic rings, the effect of spin-orbit scattering on closed electron trajectories is studied. It is shown that for one-dimensional systems within either the grand-canonical ensemble or the canonical ensemble consisting exclusively of even numbers (at zero temperature), there exist universal reduction factors due to spin-orbit scattering. No such universal factors occur in the canonical ensemble for an odd number of electrons. In higher dimensions, it is argued that strong spin-orbit scattering reduces the weak-localization contribution to the paramagnetic persistent current by a factor of 1/4. Enhancement of the paramagnetic behavior due to an additional mechanism is suggested. The relevance of these results to experiments is discussed.