We examine if the globalisation of science is accompanied by convergence in the level and structure of scientific output. We use Web of Science data on the scientific output of 205 countries for 1993, 2000, and 2008, distinguished by subject area. We found evidence of absolute and conditional β-convergence and σ-convergence in levels of scientific output, particularly after 2000. The data also show that the portfolios of the majority of the world's science systems are becoming more similar. This convergence of portfolios occurs in convergence clubs rather than as a global process. Exploratory factor analysis shows that countries cluster into eight discrete convergence clubs and perhaps only two: the `haves' and `have-nots'. Dynamic shift-share analysis reveals that growth is a normal phenomenon, output composition is only really an issue in the former Soviet Republics (negative) and the LDCs (positive) after 2000, and comparative advantages is where convergence clubs differentiate strongest. The ability of countries to improve local conditions and escape the strictures of their portfolio depends on the interplay of forces along two dimensions. between short-term dynamics and long-term stability and between the complexity of science and the predominance of national policies and institutions. Understanding the design and functioning of a science system in all its complexity is crucial to survive in a world of different speeds with intense competition and persistent gaps between rich and poor. For scientists and policy makers alike, selecting the right science portfolio and knowing the competition are key issues.