Single-layer graphene, hosting a high density of functionalized molecular-sieving atom-thick pores, is considered to be an excellent material for gas separation membranes. These functionalized atom-thick pores enable the shortest transport pathway across the membrane and competitive sorption of the target molecules over the unwanted molecules. However, incorporating high-density gas-sieving pores with sub-angstrom resolution and functionalization of heteroatoms on the 2D pores have been major challenges. This dissertation focuses on the development of a gas separation single-layer graphene membrane hosting gas-sieving and functional groups rich 2D pores. The gas-sieving pores were incorporated by O3-etching chemistry assisted by mathematical modeling to predict the gas-sieving defect density and the pore size distribution. This approach enabled a well-control of the nucleation and pore expansion rate, yielding the attractive CO2 permeance of 4400 gas permeation units (GPU; 1 GPU = 3.35×1010 mol m-2 s-1 Pa-1) and CO2/N2 separation factor of 33.4. A nucleation-decoupled expansion achieved by using dilute O3 was utilized for a slow pore expansion. The high throughput graphene membranes functionalized with CO2-philic polymer achieved an attractive carbon capture performance. Graphene membranes hosting N-substituted 2D pores have been developed to promote selective and rapid transport of CO2 across membranes using competitive sorption and gas-sieving. N-substituted 2D pores, derived by a facile reaction of oxidized graphene with NH3, show a rapid and quantitatively reversible complexation of CO2 with pyridinic N by cycles of adsorption and desorption in the spectroscopy. The phenomenon is also visualized by microscopy where pores are observed occupied and empty upon adsorption and desorption, respectively. The N-substituted 2D pores exhibit strong competitive sorption for CO2, resulting in an excellent carbon capture performance, even in the dilute CO2 source. It led to a large CO2/N2 separation factor (close to 2000) while the atomic selective layer resulted in high CO2 permeance (up to 50000 GPU). Additionally, CO2-complexation of amine groups, such as -NHCOO-, -NH3+, on the pore edges resulted in a narrower electron-density gap in the 2D pores, thereby leading to effective size-sieving separation between H2/N2 and O2/N2, with a molecular sieving resolution of 0.2 Å. The resulting graphene membrane reached O2/N2 selectivity of 6.0 with a corresponding O2 permeance of 1630 GPU, and H2/N2 selectivity of 53.5 with corresponding H2 permeance of 14460 GPU. Finally, high-performance centimeter-scale membranes were achieved thanks to the uniform and scalable chemistry of gas phase O3 etching and NH3 treatment. The graphene membranes mechanically supported by Poly(1-trimethylsilyl-1-propyne) (PTMSP) on the porous support were demonstrated for carbon capture. Both CO2-philic polymer-functionalized and pyridinic-N-substituted graphene membranes show excellent CO2 capture permeance for post-combustion and dilute CO2 sources. Overall, incorporating gas-sieving 2D pores on the graphene lattice and functional heteroatoms/functional groups at 2D pores provides new directions to enhance the performance of membranes, sensors, and catalysts.