We study the implications of climate change and the associated mitigation measures for optimal monetary policy in a canonical New Keynesian model with climate externalities. Provided they are set at their socially optimal level, carbon taxes pose no trade-offs for monetary policy: it is both feasible and optimal to fully stabilize inflation and the welfare-relevant output gap. More realistically, if carbon taxes are initially suboptimal, trade-offs arise between core and climate goals. These trade-offs however are resolved overwhelmingly in favor of price stability, even in scenarios of decades-long transitions to optimal carbon taxation. This reflects the untargeted, inefficient nature of (conventional) monetary policy as a climate instrument. In a model extension with financial frictions and central bank purchases of corporate bonds, we show that green tilting of purchases is optimal and accelerates the green transition. However, its effect on CO2 emissions and global temperatures is limited by the small size of eligible bonds’ spreads.