We develop an assignment theory to analyze the volume and composition of foreign direct investment (FDI). Firms conduct FDI by either engaging in greenfield investment or in cross-border acquisitions. Cross-border acquisitions involve firms trading heterogeneous corporate assets to exploit complementarities, while greenfield FDI involves building a new plant in the foreign market. In equilibrium, greenfield FDI and cross-border acquisitions co-exist, but the composition of FDI between these modes varies with firm and country characteristics. Firms engaging in greenfield investment are systematically more efficient than those engaging in cross-border acquisitions. Furthermore, most FDI takes the form of cross-border acquisitions when factor price differences between countries are small, while greenfield investment plays a more important role for FDI from high-wage into low-wage countries. These results capture important features of the data.