With the monetary policy lower bound a re-emerging concern in some locations, we present new insights on the impact of negative policy rates. We develop a new theoretical model to match the empirical evidence on their effects. It features a heterogeneous, oligopolistic banking sector where loan pricing is determined in part by the availability of deposit funding and in part by wholesale funding. The use of non-deposit funding ensures that the bank lending channel of negative rates remains active. We explore the impact of the policy on different types of bank: high-deposit banks may experience a fall in interest margins and profitability, which can result in reduced lending. But this is more than compensated by greater lending from low-deposit banks. We embed this banking sector in an open-economy macroeconomic model, featuring exchange-rate and capital market transmission channels, which continue to work as normal when rates are negative. These non-bank channels, combined with general equilibrium effects and an active bank lending channel, mean that the transmission of negative rates is only somewhat weaker than conventional policy.