Hydrogen is viewed as the future carbon–neutral fuel, yet hydrogen storage is a key issue for developing the hydrogen economy because current storage techniques are expensive and potentially unsafe due to pressures reaching up to 700 bar. As a consequence, research has recently designed advanced hydrogen sorbents, such as metal–organic frameworks, covalent organic frameworks, porous carbon-based adsorbents, zeolite, and advanced composites, for safer hydrogen storage. Here, we review hydrogen storage with a focus on hydrogen sources and production, advanced sorbents, and machine learning. Carbon-based sorbents include graphene, fullerene, carbon nanotubes and activated carbon. We observed that storage capacities reach up to 10 wt.% for metal–organic frameworks, 6 wt.% for covalent organic frameworks, and 3–5 wt.% for porous carbon-based adsorbents. High-entropy alloys and advanced composites exhibit improved stability and hydrogen uptake. Machine learning has allowed predicting efficient storage materials.