We use the lubrication approximation to investigate the steady two-dimensional flow of a thin film of viscous fluid on the outside of a rigid circular cylinder that is rotating about its (horizontal) axis. Primarily we are concerned with the flow that ensues when fluid is supplied continuously as a ‘curtain’ from above the cylinder, so that it flows round the cylinder and eventually falls off near the bottom. This problem may be thought of as a ‘hybrid’ of the two classical problems studied by Nusselt (1916a, b) and Moffatt (1977), concerning, respectively, flow on a stationary cylinder with a prescribed supply flux, and flow on a rotating cylinder when the supply flux is zero. For all these problems there are indeterminacies in the steady lubrication solution; we present a variety of possible solutions, including both ‘full-film’ and ‘partial-film’ solutions, and solutions that involve smooth ‘jumps’ in the free-surface profile. We show, for example, that stagnation points can occur in the flow, that solutions exist that do not have top-to-bottom symmetry, that in curtain flows the curtain generally takes a characteristic ‘buckled’ shape, and that in full-film curtain flows there is always some fluid that is ‘trapped’ near the rotating cylinder, never escaping as part of the curtain that detaches at the bottom of the cylinder. Also we show that finite-thickness films involving jumps cannot occur in these coating flows (though they are known to occur in rimming flows).