In this article, we discuss the recent evolutions of science and engineering doctoral and postdoctoral education in Europe. Indeed, Ph.Ds are crucial to the conduct of research and innovation in the national innovation systems, as they provide a large amount of input into creating the competitive advantage, notably through basic research. First, we show that Asia, and notably China, is producing more Ph.Ds than the United States and Europe. In many EU countries, the number of Ph.Ds has levelled off or even declined recently in many natural sciences and engineering fields. Second, we discuss the European situation in the international competition for talents. We study the European brain drain question, mainly at the doctoral and postdoctoral level. We find that there is an asymmetry in the flows of Ph.D students and postdoctorates between Europe and the United States, at the advantage of this latter country. These two points--production of Ph.Ds, international flows of doctorates and postdoctorates--lead us to be concerned about the future growth and innovation in Europe. In conclusion, we outline some European policy responses in the perspective of building the European Research Area and the European Higher Education Area.