Individual banks differ substantially in their foreign operations. This paper introduces heterogeneous banks into a general equilibrium framework of banking across borders to explain the documented variation. While the model matches existing micro and macro evidence, novel and unexplored predictions of the theory are also strongly supported by the data: The efficiency of the least efficient bank active in a host country increases the greater the impediments to banking across borders and the efficiency of the banking sector in the host country. There is also evidence of a tradeoff between proximity and fixed costs in banking. Banks hold more assets and liabilities in foreign affiliates relative to cross-border positions if the target country is further away and the cost of foreign direct investment is low. These results suggest that fixed costs play a crucial role in the foreign activities of banks.