Many of today's nanotechnologies and processes rely on highly confined ($<30$ nm) polymer glasses and polymer nanocomposites. These materials have aided the development of smaller and faster computer chips, improved energy storage devices, protective coatings, and next generation materials with enhanced mechanical, electrical, and optical properties. However, confinement of these materials introduces interfaces, which may drastically change fundamental properties of the material. For example, confinement of polymer glasses has shown to cause changes in the elastic modulus, which in turn may lead to significant adverse effects on the end-use applications of these materials. Understanding how confinement changes the properties of the material may provide us with a mechanism to control it, enabling the production of better materials on the nano scale. Fortunately, the length scales where confinement has an effect are reasonable to model using computer simulations. This Thesis investigates the confinement of polymer glasses and polymer nanocomposites using molecular dynamics and Monte Carlo simulations. In Chapter 2, we use a coarse-grained model to investigate the effect of backbone rigidity and confinement on polymer glasses in bulk and free-standing thin films. We extend this model throughout the entire Thesis. In Chapters 3-4, we study the process of physical aging and its role on local mobility in thin films, and, for the first time, we find a bulk property which may control important dynamical length scales in the film. In Chapter 5, we investigate the mechanical properties of confined model nanopillars, which have not been previously simulated in the context of polymer glasses, and develop close phenomenological connections between polymer glasses and metallic glasses by looking at the local non-affine strain. Lastly, in Chapter 6, we turn to polymer nanocomposites, and study the mechanism by which polymers infiltrate a highly packed nanoparticle film, a method that has been recently discovered to produce polymer nanocomposite films with very high loadings. By investigating all of these systems on the molecular level, we explain some of the underlying physics, and our results shed light on several controversies present in the literature.