There is growing evidence that over the past few decades competition across markets in the EU may have weakened: industry concentration and markups appear to have increased, while the gap between market leaders and followers seems to have widened and business dynamism seems to have declined. Against this background, this report investigates four important aspects of the state of competition in the EU. First, six sectoral cross-country price-concentration studies provide qualitative and, for mobile telecoms and airlines, empirical evidence that higher concentration seems to be associated with higher prices. Secondly, an analysis of the evolution of ‘Global Superstars’ (i.e. the most profitable of the world’s largest firms) finds that their profit rates have increased significantly over the last 25 years, and that the distribution of profits has become more skewed. We also study how Global Superstars in the IT, pharma and consumer goods sectors are protected by barriers to entry. Thirdly, a survey of EU-based exporting firms suggests that effective domestic competition within the Single Market (i) is an important driver of their global export competitiveness (in particular effective competition in upstream goods markets) and (ii) is for a majority of respondents not constraining their scale in a way which would prevent them from being successful on global export markets. Finally, relying on own estimates of markups for 117,000 firms from 23 EU Member States and a general equilibrium macroeconomic model, we estimate based on three simulation scenarios that more effective competition across markets in the EU would likely contribute significantly to more investment, employment, productivity and increase GDP by more than 2% and 4% after 5 and 10 years respectively.