The formation and stability of solids and molecules is not possible without chemical bonds, which are divided into covalent, ionic, metallic, and van der Waals bonds. A special type of intermolecular bond is hydrogen bonding, which plays a crucial role for chemical, biological, and physical processes. However, hydrogen shows a far more complex behavior when it is present in solids. In this paper, it is shown that the chemical bonding of hydrogen atoms and molecules extends far beyond the simple picture of conventional, ionic, covalent, and multicenter bonds. The interaction of H with its host material is particularly important for hydrogen storage in metallic materials such as