The building construction industry is the largest anthropogenic source of pollution, with massive energy consumption and substantial CO2 emissions. Lightweight tension structures allow the simultaneous implementation of several sustainable strategies by using recyclable low-carbon structural membranes offering a greener alternative to glass and other cladding materials. Their efficient structural load-bearing mechanisms result in significant weight savings in buildings and a drastic reduction of the environmental impact associated with material production, transportation, use, and disposal. A subgroup of lightweight materials, structural fabrics, and foils has been gaining popularity among designers and architects in recent years because of their desirable features such as high stiffness, strength, ductility, durability, and functional properties. While these structural membranes open new crucial perspectives for the clean energy transition and have been recently employed worldwide, their full potential is still limited by the lack of construction codes, advanced optimization tools, and comprehensive viscous-thermo-mechanical constitutive models. This chapter aims to foster the design of membrane structures by presenting their basic principles and recent advancements in the field. It covers the design approaches, employed materials and efforts in their characterization and modeling, implications on the sustainability of the built environment, current challenges, and future pathways from both academic research and engineering design viewpoints.