Sustainable energy supply and environmental protection are the major global scientific challenges in the 21st century, such as greenhouse gas capture, natural gas production, desalination of seawater for clean water production. Membrane separation technology offers attractive energy-efficient and environmental-friendly solutions to these challenges. This PhD thesis is focused on design and fabrication of membranes from novel molecularly defined polymers and understanding their physical properties, particularly the transport properties of gas molecules in polymer membranes. First, we demonstrate a simple approach of fabricating novel polymer nanocomposite membranes using established colloidal science. Crystalline microporous zeolitic imidazolate frameworks (ZIFs) nanocrystals are incorporated into a polyimide polymer matrix via solution mixing. The resulting nanocomposite membranes show excellent dispersion of nanoparticles, good adhesion at the interface, and enhanced gas permeability while the selectivity remain at high level. We then fabricated membranes from novel microporous polymers, polymers of intrinsic microporosity (PIMs). Using the PIM-1 polymer as a prototype, we discovered that ultraviolet irradiation of PIM-1 membranes in the presence of oxygen induces oxidative chain scission at the surface, resulting in local densification and structural transformation of free volume elements. Consequently, the membrane become asymmetric with a more gas-selective layer formed at the surface, while the overall permeability maintains at high level. Finally, we report a simple thermal oxidative crosslinking method to tailor the architecture of channels and free volume elements in PIM-1 polymer membrane by heat treatment in the presence of trace amounts of oxygen molecules. The resulting covalently crosslinked polymer networks offer superior thermal stability, chemical stability, reasonable mechanical strength, and enhanced rigidity. Most important of all, thermally crosslinked PIM-1 polymer membranes show significantly enhanced molecular sieving functions that yield remarkably high selectivity and high gas permeability, which surpass the upper bound that has been limiting the polymer membranes for decades. We also demonstrate that the thermal crosslinking method is effective for crosslinking of nanocomposite membranes with porous or nonporous fillers. These microporous molecular sieve membranes are promising for a wide range of molecular-level separation applications.