We report molecular dynamics simulation results for the equilibrium properties of polymer thin films adsorbed onto flat and curved substrates. We first systematically determined the contact angle of polymer droplets on flat substrates as a function of the substrate-monomer adsorption strength and degree of polymerization. Focusing on the fully wetted regime, we then investigate the effect of the substrate’s topography on the polymer film configuration by using substrates with varying local curvature. We find that polymer chains close to the substrate are significantly stretched (compressed) in regions of high negative (positive) curvature. Further, we find partial dewetting near regions of high curvature for thin polymer films. For thicker films, the polymers fully cover the substrate, and the shape of the liquid-vapor interface largely follows the shape of the substrate. As the film thickness is increased further, the liquid-vapor interface becomes completely flat. These simulation results are corroborated by analytical calculations using a simplified continuum model that treats the adsorbed polymers as an incompressible medium. Our findings show how the substrate topography influences the conformation of individual chains as well as the shape of the entire film.