Multi-stage production is widely recognized as an important feature of the modern global economy. This feature has been incorporated into many state-of-the-art quantitative trade models, and has been shown to deliver significant additional gains from international trade. Meanwhile, specialization across stages of production, or “vertical specialization," has been largely ignored in these models. In this paper, I provide evidence that vertical specialization is a salient feature in the international trade data, which implies that the assumption made in standard models is inaccurate. I then develop a model with multi-stage production where country-level productivity differences provide a basis for vertical specialization and additional global gains from trade beyond those currently accounted for in standard models. I quantify the gains from vertical specialization according to the model. Despite the importance of vertical specialization in the data, I find that the average gains from trade are only slightly higher than the gains suggested by standard models with multi-stage production. Moreover, much of the impact of across-stage specialization is largely offset by across-sector intermediate input linkages. These results suggest that vertical specialization is not the source of missing gains from trade that have recently confounded trade economists.