We provide a tractable theory of innovation and diffusion of technologies to explore the role of international trade and foreign direct investment (FDI). We model innovation and diffusion as a process involving the combination of new ideas with insights from other industries or countries. We provide conditions under which each country's equilibrium frontier of knowledge converges to a Frechet distribution, and derive a system of differential equations describing the evolution of the scale parameters of these distributions, i.e., countries' stocks of knowledge. In particular, the growth of a country's stock of knowledge depends only on the its trade and FDI shares and the stocks of knowledge of its trading partners. We use this framework to quantify the dynamic gains from trade in the short and long run. We also explore the model's potential to account for the post-war growth of South Korea.