Coating non-planar discrete objects is an essential manufacturing step for a wide range of products including biomedical devices, automotive parts, and confectioneries. The coating process typically involves depositing a liquid layer on the object's surface to provide specific functionalities for the final product such as biocompatibility, durability, surface protection, and aesthetic appeal. However, understanding the evolution of coatings on such objects is challenging due to the intricate shapes of some objects and the multitude of competing forces that govern the coating flow. An important aspect of the coating of discrete objects is the concurrent coating of multiple layers, which introduces additional complexity that alters the interplay of viscous, surface-tension, centrifugal, and gravitational forces within the layers. Additionally, the rheology of coating liquids is sometimes non-Newtonian, which further adds to the complexity of the problem by altering viscous forces, thereby impacting coating quality. In our first project, multilayer coating on discrete objects is studied using a model problem involving bilayer thin liquid films of immiscible Newtonian liquids on rotating cylinders. The lubrication approximation is applied to derive two coupled nonlinear evolution equations describing the heights of the two layers as a function of time and the angular coordinate. In the limit of rapidly rotating cylinders, gravitational effects are negligible, and linear stability analysis and nonlinear simulations demonstrate that a more viscous, thicker inner film or higher liquid-liquid interfacial tension suppresses instabilities driven by centrifugal forces for fixed properties of the outer layer. When gravitational effects are significant, a parametric study reveals that the critical rotation rate required to cause motion of liquid lobes that form due to gravitational drainage is lowered for a more viscous and thicker inner film due to an increase in viscous forces. These properties of the inner layer also lead to a reduction in the amplitude of temporal oscillations in the film thickness. In contrast, the stabilizing effects of liquid-liquid interfacial tension are negligible near the critical rotation rate due to the largely uniform curvature of the inner film. Results from the lubrication-theory-based model are complemented by finite-element simulations of the full two-dimensional equations, which reveal that the lubrication model works well for thicker films provided that gravitational effects are sufficiently small. In addition to advancing fundamental understanding, the results suggest strategies for improving the uniformity of coatings on discrete objects. In our second project, coating of non-Newtonian liquids on discrete objects is studied by considering a model problem involving the flow of power-law liquids on rotating cylinders. To isolate the effects of non-Newtonian rheology, we focus on the simpler case of single-layer coatings. Using the lubrication approximation, a semi-analytical expression is derived for the critical rotation rate above which the liquid film does not drain. The critical rotation rate is found to increase for shear-thinning liquids and decrease for shear-thickening liquids compared to a Newtonian liquid with the same flow consistency index, due to weaker and stronger viscous forces, respectively. To understand the temporal evolution of the film along with surface-tension effects, the full two-dimensional equations are solved using finite element simulations. At the critical rotation rate of a Newtonian liquid, a shear-thinning liquid drains, whereas the lobe of the shear-thickening liquid is found to rotate with a smaller oscillation amplitude compared to the Newtonian liquid. The predictions from the lubrication-theory-based model are found to be in good agreement with those obtained from the full two-dimensional model. Flow visualization experiments with shear-thinning liquids reveal that shear thinning leads to thinner entrained films and a wider range of rotation rates over which a smooth coating is obtained, relative to Newtonian liquids of the same zero-shear viscosity. Critical rotation rates predicted by the models agree well with the experimental observations. Overall, these studies aim to enhance our fundamental understanding of coating film evolution for multilayer and non-Newtonian liquids on a simple cylindrical geometry, and lay a foundation for systematic studies of coating on objects having more complex geometries.