Economic convergence has been one of the explicit goals of the EU from its very beginning. The prospect of higher living standards has undeniably been a major attraction of EU membership. Conversely, economic divergence may undermine support for the European project and complicate the common monetary policy in the euro area. In this policy note, we first summarise the key findings of an analysis of national and regional convergence across the EU. In particular, we show that initially poorer European countries and regions have, in general and over the longer term, made progress in catching-up with the income levels of their richer peers, even though convergence has not been a smooth process. The relative performance of countries and regions in the EU is also illustrated. We then shed some light on the extent to which the Covid-19 crisis and the ensuing recovery might impact the functioning of the EU “convergence machine”, before drawing some implications in terms of economic policy.