The article aims at presenting the experience of using the student engagement approach, which was developed on the basis of ideas of A. Astin, R. Pace, G. Kuh, P. Ewell, E. Pascarella, P. Terenzini, to assess the quality of education in Russian universities. This approach was originally implemented in the US higher education system at the end of the XX century. Due to its simplicity and relatively low costs demanded, as well as the possibility to develop effective specific recommendations for improving the educational process, this approach came to be further applied in other countries by single universities and even at national level. However, its applicability in the systems different from the US higher education is still questionable. The article presents the basic assumptions of this approach, the history of its development and spread across the world, and its potential for assessing the quality of higher education. The author pays particular attention to the implementation of the approach in Russian universities and its limitations due to peculiarities of national context. This paper is the first to summarize the results of Russian research on student engagement. It becomes possible to make certain conclusions about the applicability of this approach to the Russian system of higher education. Moreover, the author draws conclusions on the potential to compare student engagement data collected during different time intervals in different institutions and countries. Finally, the article gives practical recommendations on how to use this approach when solving different university problems. The paper is intended for university administrators to develop effective educational policy, for professors to improve teaching practices and student-faculty interaction, for researchers to understand the landscape of studies on student engagement and to distinguish promising directions for future investigations.