We propose a novel theory of banks’ liquidity management and financial fragility. Banks hold liquidity and an illiquid productive asset, thereby engaging in maturity transformation, and insure their depositors against idiosyncratic and aggregate shocks. However, strategic complementarities in the depositors’ withdrawal decisions might trigger a self-fulfilling run, modelled as a “global game”. In equilibrium, if the liquidation of the productive asset is sufficiently costly and the depositors are sufficiently risk averse, banks manage their liquidity needs during runs following an endogenous pecking order: they first deplete liquidity, and then liquidate the productive asset. Thus, under these conditions banks subject to runs are first illiquid but solvent, and then become insolvent. Ex ante, if the probability of the idiosyncratic shock is sufficiently large, banks hoard liquidity, and narrow banking is not viable.