Hydrogen is widely regarded as a key energy carrier of the future; however, its efficient and safe storage remains one of the central scientific and technological challenges. This article systematizes contemporary approaches to hydrogen storage, with a particular emphasis on solid-state materials as the most promising direction. The main storage technologies—compressed, liquid, and solid hydrogen—are reviewed, and there is an analysis of their advantages and limitations. Special attention is given to the fundamental aspects of solid-state hydrogen storage, including the thermodynamics and kinetics of hydrogen absorption and desorption processes. A comparative analysis of various material classes is presented, including intermetallic metal hydrides based on AB₅-type and AB₂-type compositions, magnesium-based systems, palladium-modified structures, as well as complex hydrides (alanates, borohydrides, and amide–imide systems).The role of nanostructuring and carbon-based nanomaterials in enhancing storage capacity and improving kinetics is also examined. Key challenges are discussed, including high desorption temperatures, slow kinetics, and material degradation. Modern strategies for performance enhancement, such as catalytic activation, alloying, and nanostructuring, are highlighted.