This article provides a characterization of the internationalization of "global" European universities and discusses the role of the State in promoting greater internationalization and competitiveness levels of prominent national universities. The analysis supports previous arguments stating that global ranking of universities is strongly based on research, but reveals that the internationalization of research universities' student population is also arranged to enhance research capacity. This finding is further reinforced by a positive association between the internationalization of the academic staff and the internationalization of the student population in one of those universities, being this association particularly strong with the doctoral student population. Finally, based on the analysis of two prominent national universities with different global competitiveness levels, we discuss the role of the State as a central supporter of these universities internationalization and global competitiveness arguing that public funding and support is critical if countries want to have national prominent universities competing at global level.