In the wake of Brexit and the Trump tariff war, central banks have had to reconsider the role of monetary policy in managing the economic effects of tariff shocks, which may induce a slowdown while raising inflation. This paper studies the optimal monetary policy responses using a New Keynesian model that includes elements from the trade literature, including global value chains in production, firm dynamics, and comparative advantage between two traded sectors. We find that, in response to a symmetric tariff war, the optimal policy response is generally expansionary: central banks stabilize the output gap at the expense of further aggravating short-run inflation---contrary to the prescription of the standard Taylor rule. In response to a tariff imposed unilaterally by a trading partner, it is optimal to engineer currency depreciation up to offsetting the effects of tariffs on relative prices, without completely redressing the effects of the tariff on the broader set of macroeconomic aggregates.