Bauxite residue, known as red mud, is a by-product of alumina production using the Bayer process. Currently, its total global storage amounts to over 4.6 billion tons, including about 600 million tons in Russia. The total global storage of red mud occupies large areas, leading to environmental damage and increasing environmental risks. Moreover, it contains a significant amount of sodium, which is easily soluble in subsoil water; therefore, a sustainable approach for comprehensive recycling of red mud is necessary. The bauxite residue contains valuable elements, such as aluminum, titanium, and scandium, which can be recovered using liquid media. In recent years, many methods of recovery of these elements from this waste have been proposed. This paper provides a critical review of hydrometallurgical, solvometallurgical, and complex methods for the recovery of valuable components from red mud, namely, aluminum, titanium, sodium, and rare and rare-earth elements. These methods include leaching using alkaline or acid solutions, ionic liquids, and biological organisms, in addition to red mud leaching solutions by extraction and sorption methods. Advantages and disadvantages of these processes in terms of their environmental impact are discussed.