This study examines the development of the productive structures of the EU using international trade data and methods from complexity theory referred to as the "product space" approach. The results show that the set of products for which a country has already a comparative advantage in international trade is a strong predictor for the type of products in which it will develop a comparative advantage and obtain significant world market shares. This implies that the development of the productive structures of a country is a highly cumulative process and any upgrading is necessarily deeply rooted in current capabilities and industrial specialisation. Complementary factors and competencies have to be built up. This makes it more difficult for countries to change their productive structures. In the light of the results of this study the smart specialisation strategy which the European Commission pursues in its cohesion policy for the years 2014-2020 seems to be well placed to foster the competitiveness of the European Union in general and the European regions in particular. Some caveats however apply.