The global energy market is shifting toward renewable, sustainable, and low-carbon hydrogen energy due to global environmental issues, such as rising carbon dioxide emissions, climate change, and global warming. Currently, a majority of hydrogen demands are achieved by steam methane reforming and other conventional processes, which, again, are very carbon-intensive methods, and the hydrogen produced by them needs to be purified prior to their application. Hence, researchers are continuously endeavoring to develop sustainable and efficient methods for hydrogen generation and purification. Membrane-based gas-separation technologies were proven to be more efficient than conventional technologies. This review explores the transition from conventional separation techniques, such as pressure swing adsorption and cryogenic distillation, to advanced membrane-based technologies with high selectivity and efficiency for hydrogen purification. Major emphasis is placed on various membrane materials and their corresponding membrane performance. First, we discuss various metal membranes, including dense, alloyed, and amorphous metal membranes, which exhibit high hydrogen solubility and selectivity. Further, various inorganic membranes, such as zeolites, silica, and CMSMs, are also discussed. Major emphasis is placed on the development of polymeric materials and membranes for the selective separation of hydrogen from CH4, CO2, and N2. In addition, cutting-edge mixed-matrix membranes are also delineated, which involve the incorporation of inorganic fillers to improve performance. This review provides a comprehensive overview of advancements in gas-separation membranes and membrane materials in terms of hydrogen selectivity, permeability, and durability in practical applications. By analyzing various conventional and advanced technologies, this review provides a comprehensive material perspective on hydrogen separation membranes, thereby endorsing hydrogen energy for a sustainable future.