The flow of liquid films on discrete objects is encountered in coating processes for a wide range of products such as biomedical devices, automobiles, and food. Describing the shape of liquid films as they flow over discrete objects is a challenging task due to the large number of forces at play. These include gravitational, inertial, viscous, surface-tension, and centrifugal forces, and the complex interplay among them may lead to the growth of instabilities that degrade the quality of the final product. Motivated by the need to improve fundamental understanding of coating flows on discrete objects, we pick cylinders that rotate about their horizontal axes as model discrete objects and investigate four model problems highly relevant to industrial coating processes for rotating discrete objects. In each model problem, the interplay among all the forces is systematically examined to reveal the critical conditions for which a smooth coating can be obtained. For coating of surfactant-laden liquids on rotating cylinders, we applied lubrication theory to derive coupled nonlinear evolution equations to describe the variation of the film thickness and surfactant concentration as a function of time, the angular coordinate, and the axial coordinate. In the absence of gravitational effects, linear stability analysis reveals that surfactant-induced Marangoni stresses suppress the growth rate of instabilities driven by centrifugal effects and hinder the leveling of perturbations to the film thickness in both the angular and axial directions. When gravitational effects are present, Marangoni stresses lower the critical rotation rate needed to cause a liquid lobe to form and rotate in the angular direction. These stresses also lead to faster damping of this lobe, giving rise to a more axisymmetric coating. With the growth of axial instabilities at long times, Marangoni stresses significantly weaken the stabilizing effect of surface-tension forces, which are found to be responsible for keeping the coating axially uniform in a stable speed window. In addition, Marangoni stresses tend to reduce the spacing between droplets that form at low rotation rates, and suppress the growth rate of rings that form at high rotation grates. Flow visualization experiments yield observations that are qualitatively consistent with our simulation results. For cylinders with complex surface geometries (i.e., topographically patterned cylinders and elliptical cylinders), the Galerkin finite-element method is used to solve the Stokes equations, augmented with a term accounting for centrifugal forces, in a rotating frame of reference. For rapidly rotating cylinders where gravitational forces are negligible, surface-tension forces tend to drive liquid to the low-surface-curvature areas (e.g., pattern troughs) leading to the formation of liquid pools, while centrifugal forces tend to drive liquid in the opposite direction, giving rise to liquid droplets. The number of droplets or pools at steady state depends on the rotation rate, strength of surface tension, pattern frequency, and cylinder aspect ratio. When gravitational forces become significant, it is possible to obtain a coating that closely conforms to the cylinder surface in the patterned-cylinder case. With an increase in the pattern amplitude, recirculation regions start to form inside the troughs, which may strongly influence mixing, mass transport, and heat transport. These reciprocation regions can appear and vanish as the cylinder rotates due to the variation of gravitational forces around the cylinder surface. In the elliptical-cylinder case, simulation results show that smaller aspect ratio corresponds to less liquid that can be supported on the cylinder and also larger gradients in film thickness. A suitably chosen time-dependent rotation rate can greatly improve coating smoothness relative to the constant-rotation-rate case. For cylinders with sufficiently small aspect ratio, film rupture and liquid shedding may occur over the cylinder tips, so simultaneous drying and rotation along with the introduction of Marangoni stresses will likely be especially important for obtaining a smooth coating.