Students from low socio-economic status (SES) or ethnic minority backgrounds are less likely to enrol into elite universities than high SES or White students. We use student-level university application data from the UK centralised university admissions service to show that these gaps cannot be accounted for by prior educational performance or subject preferences. The reasons why these gaps emerge differ according to the demographic group considered. SES gaps are predominantly driven by students' application decisions, while ethnic gaps result from minority groups' lower propensity to achieve the conditions of offers. There is widespread concern about the potential impact of COVID-19 and the resulting alternative assessment arrangements for the 2020 application cohort. We find no evidence of a differential impact on access to elite institutions, but show that Black and South Asian students, who in normal years are successful at finding a place at later stages of the application process were more likely to be squeezed out.