The transition to a sustainable economy requires a more effective and less energy-intensive industry. Membrane technology could augment or partially substitute classical molecular separation processes such as distillation to reduce energy, carbon, and water intensity. Organic solvent nanofiltration and reverse osmosis (OSN and OSRO) can positively impact the petrochemical, pharmaceutical, food, and fine chemical industries, among others, if broadly implemented. While hybrid and inorganic materials have the potential for game-changing performance, polymeric membranes provide key advantages in scalability and processability. Improved materials able to operate in challenging conditions, including combinations of organic solvents, high temperatures, extreme pHs, and oxidative environments are crucial. This is a comprehensive review of the state-of-the-art of polymeric membranes for use in OSN and OSRO, including a critical analysis of current academic approaches and potential polymer systems capable of enabling high-temperature liquid phase membrane separations. The challenges and prospects of OSN and OSRO membranes are discussed in the final section.