This paper develops a model of banking across borders where banks differ in their efficiencies that can replicate key patterns in the data. More efficient banks are more likely to have assets, liabilities and affiliates abroad and have larger foreign operations. Banks are more likely to be active in countries that have less efficient domestic banks, are bigger and more open to foreign entry. In the model, banking sector integration leads to bank exit and entry and convergence in the return on loans and funding costs across countries. Bank heterogeneity matters for the associated welfare gains. Results suggest that differences in bank efficiencies across countries drive banking across borders, that fixed costs are crucial for foreign bank operations and that globalization makes larger banks even larger.