This thesis sheds light on the motives, the nature and the implications of banking across borders. In Chapter 1, co-authored with Tim Schmidt-Eisenlohr, I examine the challenges that increased financial integration presents for policy cooperation as financial crises and government intervention have stronger effects beyond borders. We provide a model of international contagion allowing for bank bailouts. While a social planner trades off  tax distortions, liquidation losses and intra- and inter-country income inequality, in the noncooperative game between governments there are inefficiencies due to externalities, a lack of burden sharing and free-riding. We show that, in absence of cooperation, stronger interbank linkages make government interests diverge, whereas cross-border asset holdings tend to align them. We analyze different forms of cooperation and their effects on global and national welfare. In Chapter 2, I show that first principles of international trade theory go far in explaining banking across borders. I develop and test a theoretical model where trade in banking services arises from differences in relative factor endowments and in banking technology across countries. The analysis reveals that differences in endowments lead to international banking where banks raise capital in the home market and lend it abroad. In contrast, differences in banking sector efficiency make banks intermediate capital locally in the foreign market, an activity which is denoted as global banking. The foreign assets and liabilities of a banking sector reflect the importance of each of the two driving forces. Key model predictions regarding the cross-country pattern of foreign banks asset and liability holdings are strongly supported by the data. In Chapter 3, I develop a general equilibrium model that can explain heterogeneity in banks' international and global activities across countries, across banks, and over time that is consistent with empirical facts. Choosing between investing and raising deposits at home and abroad, banks sort endogenously into cross-border lending and FDI: more efficient larger banks are more likely to engage in both of these activities (extensive margin). At the same time, they hold more foreign assets and liabilities (intensive margin). The model predicts precisely how the intensive and extensive margins change as capital accounts and banking sectors become more integrated, and how they vary with recipient and source country characteristics. It shows that capital flows depend crucially on banking sector efficiency in the source and the recipient country.