High level nuclear waste from reprocessing of spent nuclear fuel has to be solidified in a stable matrix for safe long-time storage. Vitrification in borosilicate glasses is the technique accepted worldwide as the best combination of engineering constraints from fabrication and physicochemical properties of the matrix. A number of different glasses was developed in different national programs. The criteria and the reasons for selecting the final compositions are described briefly. Emphasis is placed on the French product R7T7 and on thermal and physicochemical properties though glasses developed in other national projects (e.g., the German product GP 98/12, etc.) are also treated. The basic physical and mechanical properties and the chemical durability of the glass in contact with water are described. The basic mechanisms of aqueous corrosion are discussed and the evolving modelling of the leaching process is dealt with, as well as effects of container material, backfill, etc. The thermal behavior has also been studied and extensive data exist on diffusion of glass constituents (Na) and of interesting elements of the waste such as the alkalis Rb and Cs or the actinides U and Pu, as well as on crystallization processes in the glass during storage at elevated temperatures. Emphasis is placed on the radiation stability of the glasses, based on extensive studies using short-lived actinides (e g., Cm-244) or ion implantation to produce the damage expected during long storage at an accelerated rate. The radiation stability is shown to be very good, if realistic damage conditions are used. The knowledge accumulated in the past years is used to evaluate and predict the long-term evolution of the glass under storage conditions.