Pacific Northwest Laboratory (PNL) has been investigating advanced waste forms, including matrix waste forms, that may provide a very high degree of stability under the most severe repository conditions. The purpose of this study was to recommend practical matrix materials for future development that most enhance the stability of the matrix waste forms. The functions of the matrix were reviewed. Desirable matrix material properties were discussed and listed relative to the matrix functions. Potential matrix materials were discussed and recommendations were made for future matrix development. Th1e matrix mechanically contains waste cores, reduces waste form temperatures, and is capable of providing a high-quality barrier to leach waters. High-quality barrier matrices that separate and individually encapsulate the waste cores are fabricated by powder fabrication methods, such as sintering, hot pressing, and hot isostatic pressing. Viable barrier materials are impermeable, extremely corrosion resistant, and mechanically strong. Three material classes potentially satisfy the requirements for a barrier matrix and are recommended for development: titanium, glass, and graphite. Polymers appear to be marginally adequate, and a more thorough engineering assessment of their potential should be made.