The coating of discrete objects having surface topography is an important step in the manufacturing of a broad variety of products. To develop a fundamental understanding of this problem, we study liquid-film flow on rotating cylinders patterned with sinusoidal topographical features. The Stokes equations, augmented with a term accounting for centrifugal forces, are solved in a rotating reference frame using the Galerkin finite-element method (GFEM). A nonlinear evolution equation for the film thickness based on lubrication theory is also solved numerically and its predictions are compared to those from the GFEM calculations. When gravitational effects are negligible and the rotation rate is sufficiently low, liquid accumulates over the pattern troughs before merging to form multiple larger drops (located over troughs) whose number at steady state depends on the topography wavelength and rotation rate. When the rotation rate is sufficiently high, similar merging events occur, but liquid accumulates over the pattern crests at steady state. When gravitational forces become significant, it is possible to obtain a coating that closely conforms to the surface topography. The GFEM calculations are in agreement with predictions from the lubrication model provided the free-surface curvatures are sufficiently small. For sufficiently large pattern amplitude and film thickness, the GFEM calculations show that recirculation regions inside the troughs can appear and vanish as the cylinder rotates due to the variation of gravitational forces around the cylinder surface. This phenomenon, along with flow reversal over the crests, may strongly influence mixing, mass transport, and heat transport.