Abstract: The striking growth in the trade share of output is one of the most important developments in the world economy since World War II. Two features of this growth present challenges to the standard trade models. First, the growth is generally thought to have been generated by falling tariff barriers worldwide. But tariff barriers have decreased by only about 11 percentage points since the early 1960s; the standard models cannot explain the growth of trade without assuming counterfactually large elasticities of substitution between goods. Second, tariff declines were much larger prior to the mid 1980s than after, and yet, trade growth was smaller in the earlier period than in the later period. The standard models have difficulty generating this nonlinear feature. This paper develops a two-country dynamic Ricardian trade model that offers a resolution of these two puzzles. The key idea embedded in this model is vertical specialization, which occurs when countries specialize only in particular stages of a good’s production sequence. The model generates a nonlinear trade response to tariff reductions and can explain over 50 percent of the growth of trade. Finally, the model has important implications for the gains from trade.