The article examines the global university reputation race, launched in 2003. Between 2003 and 2010, there appeared a cluster of publications on the qualitative comparative analysis of their methodologies, and since 2010, a cluster of publications on the quantitative comparative analysis of university rankings has started to form. The review made it possible to identify a number of unsolved problems concerning the stability of university rankings, aggregation of the number of universities and their Overall Scores (Total Scores) by country in various rankings. Our study aimed at solving these tasks was carried out for TOP-100s of ARWU, QS, and THE rankings. When calculating the fluctuation range of the university rankings, the top twenty of the most stable and most unstable university rankings were identified in the rankings under study. The best values of the aggregated indicators by the number of universities and the Overall Scores were identified for the USA and the UK.