High-level radioactive waste is accumulating at temporary storage locations around the world and will eventually be placed in deep geological repositories. The waste forms and containers will be constructed from glass, crystalline ceramic, and metallic materials, which will eventually come into contact with water, considering that the period of performance required to allow sufficient decay of dangerous radionuclides is on the order of 105-106 years. Corrosion of the containers and waste forms in the aqueous repository environment is therefore a concern. This Review describes the recent advances of the field of materials corrosion that are relevant to fundamental materials science issues associated with the long-term performance assessment and the design of materials with improved performance, where performance is defined as resistance to aqueous corrosion. Glass, crystalline ceramics, and metals are discussed separately, and the near-field interactions of these different material classes are also briefly addressed. Finally, recommendations for future directions of study are provided.