Complementing Europe's bank-based system with deeper capital markets and more cross-border financial integration promises benefits, but despite long-running debate and policy action, financial system change remains slow. Fintech has the potential to change financial intermediation structures substantially. It could disrupt existing financial intermediation with new business models empowered by intelligent algorithms, big data, cloud computing and artificial intelligence. Lower costs and potentially better consumer experiences could be the driving forces. However, empirically, fintech remains at a small scale, especially in the European Union. Even the largest fintech market, in China, is of marginal size compared to overall financial intermediation. In the EU, much of fintech is concentrated in the United Kingdom. We argue that policymakers need to consider four questions urgently: (1) Develop a European or national fintech market? (2) What regulatory framework to pursue? (3) Should supervision of fintech be exercised at the European level? (4) What is the overall vision for the EU's financial system? Getting the answers to these questions right at an early stage of market development would be an opportunity to shape a stable and cost-efficient financial system. In contrast, late action could mean that Europe loses out to foreign competitors and misses an opportunity to improve financial intermediation in Europe.