Discontinuous molecular dynamics simulations of a model polymer have been conducted to investigate the glass transition of ultrathin films and the mechanical properties of nanoscopic structures. Continuum mechanics models have been applied to interpret simulation data and extract apparent Young's Moduli. Consistent with experiments, the results of simulations indicate that the glass transition temperature of thin films can be higher or lower than that of the bulk, depending on the nature of polymer-substrate interactions. Simulations also indicate that the mechanical properties of nanoscopic structures can be considerably different from those of the bulk. An analysis of molecular strain distributions in nanostructures undergoing a deformation indicate that significant stress relaxation occurs at air-polymer interfaces. A comparison of these distributions to the results of continuum, finite-element calculations reveal pronounced differences between the continuum and molecular approaches.