By varying the number of acetylenic linkages connecting aromatic rings, a new family of atomically thin graph-n-yne materials can be designed and synthesized. Generating immense scientific interest due to its structural diversity and excellent physical properties, graph-n-yne has opened new avenues toward numerous promising engineering applications, especially for separation membranes with precise pore sizes. Having these tunable pore sizes in combination with their excellent mechanical strength to withstand high pressures, free-standing graph-n-yne is theoretically posited to be an outstanding membrane material for separating or purifying mixtures of either gases or liquids, rivaling or even dramatically exceeding the capabilities of current, state-of-art separation membranes. Computational modeling and simulations play an integral role in the bottom-up design and characterization of these graph-n-yne materials. Thus, here, the state of the art in modeling α-, β-, γ-, δ-, and 6,6,12-graphyne nanosheets for synthesizing graph-2-yne materials and 3D architectures thereof is discussed. Different synthesis methods are described and a broad overview of computational characterizations of graph-n-yne's electrical, chemical, and thermal properties is provided. Furthermore, a series of in-depth computational studies that delve into the specifics of graph-n-yne's mechanical strength and porosity, which confer superior performance for separation and desalination membranes, are reviewed.