The article deals with topical issues of ensuring the competitiveness of universities and national systems of higher education, primarily Russian, in the context of globalization. Ensuring a high level of competitiveness of the education system on the world arena is based on effective strategies for the development of national universities. The implementation of these strategies should contribute to the achievement of the level of world standards in education, research and development. The most representative examples of a competitive university are the so-called “world class universities” (WCU), included in the well-known international ratings (ARWU, THE, QS) are world-class universities. The strategy of achieving a level of competitiveness by a specific university, which makes it possible not only to get and gain a foothold in the international rating, but also to rise, is directly connected with the national (state) strategy for the development of higher education. According to S. Marginson, in the context of the globalization of education in different countries of the world, three basic models of strategies for the development of world-class universities (“breadth”, “deep” and combined – “breadth and depth”) are practiced. The strategy for the development of the Russian system of higher education was initially based on a strategy “deep” (the program for supporting Russian universities “5 in 100”). However, at present, the emphasis has shifted to the strategy of promotion in the disciplinary (subject) rating. It, quite correlates with the target scenario, formulated in the “Strategy of scientific and technological development of the Russian Federation”, approved by the Decree of the President of the Russian Federation.