How should a government bail out a heterogeneous banking system subject to systemic self-fulfilling runs? To answer this question, we develop a theory of banking with multiple groups of depositors of different size and wealth, where systemic self-fulfilling runs emerge as a consequence of a global game, and a government uses a public good to bailout banks through liquidity injections. In this framework, we characterize the endogenous probability of a systemic self-fulfilling run, and the conditions under which a full bailout cannot be part of the equilibrium. The optimal bailout strategy should target those banks whose bailout has the largest marginal impact on the probability of a systemic self-fulfilling run, and whose depositors are at the lower end of the wealth distribution.