This paper builds a theory of the shape of the distribution of total-factor productivity (TFP) across countries. The data on productivity suggests vast differences across countries, and arguably even has twin peaks. The theory proposed here is consistent with vast differences in long-run productivity, and potentially also with a twin-peaks outcome, even under the assumption that all countries are ex-ante identical. It is based on the hypothesis that TFP improvements in a given country follow a Nelson-Phelps specification. Thus, they derive from past investments in the country itself and, through a spillover (or catch-up) term, from past investments in other countries. We then construct a stochastic dynamic general equilibrium model of the world which has externalities: each country invests in TFP and internalizes the dynamic effects of its own investment, while treating other countries' investments as given. Average world growth is endogenous, as is the distribution of TFP across countries. We find that small idiosyncratic TFP shocks can lead to large long-run differences in TFP levels and that, in the long run, the world distribution of TFP across countries may be asymmetric, i.e., twin-peaked, or bimodal. More specifically, twin-peaked world distributions of TFP arise if the catch-up term in the Nelson-Phelps equation has a sufficiently low weight. If, on the other hand, technological catch-up is important, the world distribution of TFP is unimodal, though it may still have large dispersion.