This review analyzes the recent literature on the direct experimental determination of molecular and atomic-level nature of adsorption sites, mechanisms of adsorption under the mild (close to ambient) conditions, desorption of sulfur-aromatic compounds, and surface chemical reactions upon regeneration of the ospento sorbents for desulfurization of the liquid fuels. Supported noble, transition metals and metal oxides, binary, and ternary metal oxides, activated carbons, zeolites, supported polymers, pervaporation membranes and other advanced adsorptive desulfurization materials are discussed. The recent trends in developing the deep and ultradeep desulfurization sorbents are discussed, and challenges of the direct determination of molecular mechanisms are described.