This thesis consists of three essays about international trade and wage inequality. Essay I characterizes optimal trade and FDI policies in a model with monopolistic competition and firm-level heterogeneity similar to Helpman et al. (2004). I find that both the optimal import tariffs and the optimal FDI subsidies discriminate against the more profitable foreign firms. This is because of the existence of a wedge between the private incentives of exporting and FDI firms, and the incentive of the representative agent. Essay II develops an elementary theory of global supply chains. It considers a world economy with an arbitrary number of countries, one factor of production, a continuum of intermediate goods, and one final good. Production of the final good is sequential and subject to mistakes. In the unique free trade equilibrium, countries with lower probabilities of making mistakes at all stages specialize in later stages of production. Using this simple theoretical framework, it offers a first look at how vertical specialization shapes the interdependence of nations. Essay III proposes a model that has as ingredients heterogeneity of workers and firms, complementarity between occupations within each firm and complementarity between workers and firms/occupations. The competitive equilibrium features positive assortative matching and leads to both within- and between- firm wage variations. Comparative static results are then derived to generate new insights about changes in these components of wage inequality.