Start-ups founded by university students and graduates play a substantial role in bringing new knowledge to the market and in employment creation, a role that appears to be even more important than that played by the typical technology transfer activities carried out by universities. We use a population-based approach to explore entrepreneurship among 61,115 graduate alumni of 64 Italian universities. In order to assess the potential supply of highly educated entrepreneurs, we develop a novel empirical approach to analyse engagement in entrepreneurship, based on the idea that entrepreneurship is a process that begins with intention and ends in action. We find that the share of intentional entrepreneurs, among recent cohorts of graduates in Italy, is large in comparison to the small share of actual entrepreneurs detected five years after graduation. We discuss which barriers may deter intentional entrepreneurs from being engaged in entrepreneurship and how universities can trigger the entrepreneurial process and close the gap between entrepreneurial intentions and action.