This paper provides an integrated analysis of multinational companies' global production and innovation. We establish novel stylized facts using rich data on the network of production affiliates and patent activity of German multinationals. We rationalize these facts with a heterogeneous-firm model, in which companies jointly determine the location and scale of production, basic innovation and applied innovation, under asymmetric complementarities across these three activities. Empirical evidence consistent with the model indicates that bigger MNCs innovate more intensively in terms of patent frequency and quality, and offshore innovation to more countries, including both countries with and without production affiliates. Moreover, MNCs' innovation portfolio follows countries' comparative advantage across technology classes, with applied innovation more likely to be co-located with production than basic innovation.