Elastomer seals are widely used in the oil and gas industry and form small but crucial parts of critical mechanical and electronic components. Ideally, to prevent damage to such components, the seals should act as impermeable barriers. However, under the high pressures and temperatures found downhole, they are liable to suffer two main types of permeation-driven failure, either by gases permeating through the entire seal, or by dissolved gases causing swelling and rupture, in a process known as explosive decompression. Neither failure mode is particularly well understood, and experimental approaches have encountered difficulties in replicating service conditions and providing insight into the mechanisms responsible. In this work, I use a molecular-simulation-based approach to investigate the drivers of and trends in the permeation of gases as a function of the underlying polymer chemistry and the environmental conditions. I focus on two elastomers in particular that are widely used in the oil and gas industry due to their resistance to hydrocarbons and to thermal degradation: nitrile butadiene rubber, or NBR, which is a statistical copolymer of acrylonitrile and butadiene, and hydrogenated-NBR, or HNBR, which is a polymer derived from the cross-linking and saturation of NBR. For NBR, I first develop a fully atomistic model using the OPLS-AA force-field. I demonstrate that solubility increases with acrylonitrile content, the cyano group plays a crucial role in the enhanced solubility of polar gases, and that the effects of pressure and temperature are heavily gas-dependent. In contrast, the diffusivity is found to decrease with increasing acrylonitrile content. For HNBR, I show that, while the diffusivity decreases with increasing cross-link fraction, counter-intuitively, the solubility increases. The work in this thesis provides a set of molecular-level design principles for the future development of corrosion- and decompression-resistant elastomers.