This paper explores the potential impacts of climate change and mitigation policies in the Euro Area, considering the uncertainty and heterogeneity in both climate and economic systems. Using the MATRIX model, a multi-sector and multi-agent macroeconomic model, we simulate various climate scenarios by employing different carbon cycle models, damage functions, and marginal abatement curves found in the literature. We find that heterogeneous climate damages amplify both the magnitude and the volatility of GDP losses associated with global warming. By the end of the century, we estimate that assuming homogeneous shocks may underestimate the effects of climate change on aggregate output by up to one-third. Moreover, we find that the speed and feasibility of a low-carbon transition crucially depend on (i) the ambition in emission reduction targets set by the policymaker, which determine the level of a carbon tax, and (ii) the rate of technological progress, which influences the shape of the abatement cost curve.