Within a polymer thin film, free-volume elements have a wide range of size and topology. This broad range of free-volume element sizes determines the ability for a polymer to perform molecular separations. Herein, the free volume and transport properties (diffusion, permeability, and selectivity) in both rubbery and glassy polymers were simulated using fully atomistic models. Extension of the computational tool to study the void structure in proteins is also included in this thesis. Six permeable thermally rearranged (TR) polymers and their precursors were studied. Using atomistic models, cavity size (free volume) distributions determined by a combination of molecular dynamics and Monte Carlo methods were consistent with experimental observation that TR polymers are more permeable than their precursors. The cavity size distributions determined by simulation were also consistent with free volume distributions determined by positron annihilation lifetime spectroscopy. The diffusion, solubility and permeation of gases in TR polymers and their precursors were also simulated at 308 K, with results that agree qualitatively with experimental data. A new hybrid Monte Carlo/Molecular Dynamics method is developed for estimating the slow diffusion processes of light gases transporting in glassy polymers. Diffusion coefficients, as small as 10⁻⁵ to 10⁻⁹ cm²/s are estimated for penetrants in four different polymers at 298 K. In all cases, agreement between literature experimental data and values obtained from the fast hybrid molecular dynamics method ranges from good to excellent. A new technique is developed using Monte Carlo methods to characterize the cavity size distribution and surface atoms in globular proteins. New statistical metrics have been defined for the structural characterization of globular proteins. Some of these metrics include volume, surface area, asymmetry ratio, interior cavity size distribution, and the identification of percolation channels. Wild-type (WT) myoglobin (Mb) and 5 Mb mutants have been studied in this research as examples. An analysis of cavity statistics provides an efficient method to quantify local properties such as packing density and transport pathways. The average cavity sizes of WT Mb and its mutants are around 4.0-5.0 Å. / text