Education is a main facilitator of social, inter-generational mobility, and high education (HE) plays a major role in this. But while returns to HE are large, they also vary widely by the characteristics of programmes, those of students and how the two align. This paper investigates the role of sorting between students and high education programmes in driving social mobility in the UK. We develop a model of education, labour supply and earnings that allows for a rich characterisation of heterogeneity in the skills that different educational programmes provide. A critical feature of our model is that students and programmes meet in a matching market in equilibrium, where the sorting of students to programmes is determined. We exploit higher education funding reforms and rich administrative data to estimate the model and show that it reproduces empirical patterns accurately. We use the model to quantify the role of high education for social mobility and to run counterfactual analysis of policies aiming at improving participation among students from poorer backgrounds. We conclude that policies targeting the demand side are often insufficient to move the needle on social mobility given capacity constraints; supply side policies could potentially be highly effective.