The successful implementation of scientific research is one of the key factors for sustainable development, including the development of tertiary education. A leading or “world-class university”, today, transfers knowledge to innovation, bearing the concept of “academic excellence”, and features of “research” and “entrepreneurial” universities highly match the SDGs. This article presents an analysis for scientific quality-assessing indicators at research universities. It also studies university science models in different countries, the features of university types—engineering (technical or technological) and comprehensive (multidisciplinary)—and the arising problems with their possible solutions for university science on the whole. The authors suggested a composition of indicators for assessing the quality of the fundamental scientific research and engineering project results and the performances of universities’ specialized scientific units. The respondents of the survey weighed the significance of the indicators. The research used methods of analysis, brainstorming, Ishikawa diagram building, and a survey of specialists. The results obtained can be useful for the improvement of quality management systems (QMSs) at research universities. Some implications of this study could be extended to improve QMS and management processes of specialized scientific organizations that belong to the state, business, and non-profit sectors of science. © 2024 by the authors.