The exposure of sovereigns to climate risks is priced and can affect credit ratings and debt servicing costs. I will discuss how integrated assessment models (IAMs) can be linked with stochastic debt sustainability analysis to inform our understanding of climate risks to sovereign debt dynamics and assess the available fiscal space to finance climate policies. The narrative scenario architecture developed within the IPCC and the transition scenarios of NGFS can be adopted 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 model ensembles, respectively, we can address these three challenges. I will discuss two applications combining well-known climate models with a high-realism debt sustainability analysis model, and discuss (i) sovereign debt effects of the transition to low-carbon economies on a large panel of countries, and (ii) effects of damages and adaptation. Results show that adaptation pays, but raises the question of who will pay for it.