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The object of this research is the activity of 25 leading research universities in the world. One of the most problematic issues is the in-depth analysis of the ranking indicators of world-class research universities in order to identify the priorities for the development of research universities in the European Union. The following scientific methods were used in the research: generalizations, analysis, synthesis, graphical method, and also comparison method. The study is based on data on the global university rankings in QS World University Rankings 2018 and empirical data from publicly available sources provided by the research universities. The article also systematizes the scientific views of the leading scientists of Europe and the world on the course of development of research universities and analyses the experience of implementing joint scientific projects in the EU countries. The importance of international cooperation between research universities in the fields of academic mobility and scientific research is underlined due to the fact that the global challenges and prospects that the research universities in the European Union and the world are facing are transient and create a tough competitive environment. Strengthening their leadership potential and developing entrepreneurial thinking has been revealed as a modern trend in the development of research universities. The article characterizes the 25 best world-class research universities, and the factors which have enabled them to uncover their strengths and systematize the key areas of in-depth specialization. The contrasting differences between the analysed research universities are shown, as there are significant disparities in the total number of students, foreign students and staff of research universities, showing fundamentally different approaches to the organization of educational and research processes, even within a single country. This is due to the fact that the ratio of student per staff representative significantly affects the quality of education. The author proposes the key development priorities for research universities in the European Union, which consist in intensified work on: academic reputation; the number of foreign students; reputation of employers; improvement of the ratio of the number of students to the number of staff; improvement of quality and increase in the number of citations of scientific publications by university researchers; increasing the share of the university's participation in international educational and scientific networks of other countries and continents. [ABSTRACT FROM AUTHOR]