Global scholarship has documented gender discrepancies in power in higher education institutions (HEIs) for several decades. That research is now supported by wider gender equality movements such as those concerned with unequal pay and sexual harassment. Underlying these is the under-representation of women in senior management and full professorial positions. Thus, for example, in the US and the EU men make up the overwhelming majority of those in senior management (Rectors/Presidents/Vice Chancellors) and in full professorial positions. Variation within and across countries and types of institutions suggests that it is necessary to go beyond explanations for this at the individual level. Drawing on research in the sociology of education; higher education studies; management and leadership studies; gender studies; science and policy studies, and using quantitative data, experimental studies, individual case studies and comparative qualitative studies of HEIs, this article focuses on three discourses which legitimate the under-representation women in senior positions namely excellence, fit and national relevance. It evaluates interventions to deal with this including unconscious bias training, mentoring, gender mainstreaming interventions such as Athena SWAN and ADVANCE and 'nudging' leaders to end gender inequality. It concludes that a fundamental transformation of HEIs is required and suggests ways forward.