The international linkages between banks play a crucial role in today's global economy. Existing models explain these links largely on the basis of portfolio theory, in which banks diversify lending. These models have found limited empirical support and do not speak to several relevant dimensions of the data. They do not explain heterogeneity in the degree to which banks operate through foreign affiliates, fund their activities abroad or matter for local lending in foreign countries. This paper proposes a complementary theory of banking across borders that is based on elements of international trade theory. In the model, banking across borders arises because countries differ in their relative factor endowments and in the efficiency of their banking sectors. Based on these differences, the pattern of foreign bank asset and liability holdings emerges endogenously. This parsimonious model provides a rationale for the observed heterogeneity in foreign bank activities and is consistent with key patterns in the data. •New model of banking across borders.•Predicts banking sector foreign assets and liabilities.•Foreign bank positions depend on differences in relative factor endowments;•and on differences in relative banking sector efficiencies across countries.•The theory is consistent with key patterns in international banking data.