An analytical investigation is carried out to determine the conditions for instability in a viscous fluid contained between rotating coaxial cylinders of arbitrary radius ratio. A solution method is outlined and then applied to cylinders having radius ratios ranging from 0·95 to 0·1. Consideration is given to both cases wherein the cylinders are rotating in the same direction and in opposite directions. Results are reported for the Taylor numbers and wave-numbers which mark the onset of instability. The present results are also employed to delineate the range of applicability of the closed-form instability predictions of Taylor and of Meksyn, which were derived for narrow-gap conditions.