The exposure of sovereigns to climate risks is priced and can affect credit ratings and debt servicing costs. I argue that the climate risks to fiscal stability are not receiving adequate attention and discuss how to remedy the situation. After providing evidence of divergent climate risks to advanced economies, I describe the transmission channels from climate change to public finance. Then, I suggest how integrated assessment models (IAMs) can be linked with stochastic debt sustainability analysis (DSA) to inform our understanding of climate risks to sovereign debt dynamics and assess the available fiscal space to finance climate policies. I argue for adopting the narrative scenario architecture developed within the IPCC to bring structure and transparency to the analysis. The analysis is complicated by deep uncertainty —risks, ambiguity, and mis-specifications— of climate change. Using scenario trees, narrative scenarios, and ensembles of models, respectively, we can deal with these three challenges. I illustrate using two prominent IAMs to generate the debt dynamics of a high-debt country under climate risks to economic growth and find adverse effects from as early as 2030.