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Bias adjustment is often a necessity in estimating climate impacts because impact models usually rely on unbiased climate information, a requirement that climate model outputs rarely fulfil. Most currently used statistical bias-adjustment methods adjust each climate variable separately, even though impacts usually depend on multiple potentially dependent variables. Human heat stress, for instance, depends on temperature and relative humidity, two variables that are often strongly correlated. Whether univariate bias-adjustment methods effectively improve estimates of impacts that depend on multiple drivers is largely unknown, and the lack of long-term impact data prevents a direct comparison between model outputs and observations for many climate-related impacts. Here we use two hazard indicators, heat stress and a simple fire risk indicator, as proxies for more sophisticated impact models. We show that univariate bias-adjustment methods such as univariate quantile mapping often cannot effectively reduce biases in multivariate hazard estimates. In some cases, it even increases biases. These cases typically occur (i) when hazards depend equally strongly on more than one climatic driver, (ii) when models exhibit biases in the dependence structure of drivers and (iii) when univariate biases are relatively small. Using a perfect model approach, we further quantify the uncertainty in bias-adjusted hazard indicators due to internal variability and show how imperfect bias adjustment can amplify this uncertainty. Both issues can be addressed successfully with a statistical bias adjustment that corrects the multivariate dependence structure in addition to the marginal distributions of the climate drivers. Our results suggest that currently many modeled climate impacts are associated with uncertainties related to the choice of bias adjustment. We conclude that in cases where impacts depend on multiple dependent climate variables these uncertainties can be reduced using statistical bias-adjustment approaches that correct the variables' multivariate dependence structure.