A system of classifying research unviersities is proposed based on quantitative criteria. Data from several studies were used to develop a list of 57 leading U.S. research universities. The Carnegie Commission's 1973 and 1976 classification of "Research Universities I" and the Academy for Educational Development's listing are presented, along with research university lists developed by the Association of Research Libraries. Additional lists include: the leading universities in terms of total financial support for research for 1983; and the leading research universities based on total research and development expenditures, Ph.D. production, faculty size, and size of research libraries. These lists, which rank universities on the basis of standards, were subjected to a four-step elimination process, which begins with the 100 leading universities that received the largest support for scientific research in 1983. From this list, schools were eliminated that did not produce at least 100 Ph.D.s in 1983, and schools that employed fewer than 1,000 full-time scientists and engineers as faculty members. Further selection was based on having an outstanding research library. (SW)