This dissertation focuses on the fundamental understanding of liquid dynamics confined in polymer membranes. Such knowledge guides the development of better polymer membranes for practical applications and contributes to the general understanding of confined liquid dynamics in various nanoporous materials. First, we investigate the membrane transport by experimental measurements on a PFSA membrane and computer modeling of the confined liquid molecules. We probe the nano-scale environment in the ionomer membrane by determining the activation energy of diffusion. We notice two structural features of the PFSA membrane that dominate membrane transport. At relatively high hydrations, the nano-scale phase-separation creates bulk-like water in the ionomer membrane and prompts fast transport of mobile species. At relatively low hydrations, the nanoconfinement of the polymer matrix leads to the ordering of confined water and prompts a high energy barrier for transport. We then delve deeper into the confinement effect by molecular modeling of various nanoconfining geometries, including carbon nanotubes, parallel graphene sheets, and parallel rigid rods. We notice retarded water dynamics under hydrophobic confinement regardless of the geometry. We further investigate the confined water by determining the residence time of water around water, which evaluates the timescale of associations between water molecules. We learn that a decreasing confinement size prompts longer associations among water molecules. Further, we propose that the prolonged associations are responsible for the retarded water dynamics under hydrophobic confinement. Next, we turn our attention to the effect of interactions between mobile species (mostly water molecules) and a confining surface. In ionomer membranes, interactions between mobile species and the ionic groups dominate the water-surface interactions. We start by looking at water-ion interactions in bulk solutions. Using solutions at varying concentrations, we notice a temperature-concentration superposition behavior from diffusion coefficients of water molecules and ions in the solutions in both experimental and computational results. Observation of this superposition behavior in bulk solutions is unprecedented. The temperature-concentration superposition parallels the well-known time-temperature superposition. We are able to extract the offset of reciprocal temperature, which fits well to a Williams-Landel-Ferry type equation. The temperature-concentration superposition points to the new perspective that the effect of ions on water dynamics can be similar to the effect of lowering temperature. We further investigate the effect of ions by modeling ions/charges onto confining geometries. Remarkably, we reveal that the presence of ions can break the ordered water structure induced by confinement. The hydrophobic confinement prompts the ordering of water molecules, which leads to slower diffusion and higher activation energy. The presence of ions/charges on the confining surface has multiple effects on the dynamics of confined water. First, the ions associate strongly with neighboring water molecules while breaking the hydrogen-bonding network between water molecules. Second, the disruption of the hydrogen-bonding network leads to decreased activation energy of diffusion and enhanced water mobility. At relatively high ion density, the water-ion interactions overcome the structure-breaking effect and lead to retarded water diffusion. Overall, the studies presented in this dissertation augment our understanding of water transport in nanostructures by revealing the rich behavior of liquid-water dynamics under both hydrophobic and ionic confinement.