This article applies different copulas to investigate the complex dependence structure between EU emission allowance (EUA) futures returns and those of other commodities, equity and energy indices. The analysis yields important insights into the relationship between carbon, commodities and financial markets. Firstly, we find a significant relationship between EUA returns and those of the other considered variables that is most appropriately modelled by a Gaussian and Student-t copula. These results contradict some earlier studies that report no statistically significant or even negative correlations between returns of emission allowances and other financial variables. Secondly, considering time-varying copulas shows that the estimated copula parameters are not constant over time. We find in particular that the dependence is stronger during the period of the financial crisis. In a Value-at-Risk (VaR) analysis, finally, we further illustrate the advantages of copula methods. In particular, the Student-t copula provides an appropriate quantification of VaR at different confidence levels while other models fail to specify the risk correctly. This analysis shows that ignoring the actual nature of dependence might lead to an underestimation of the risk for portfolios combining EUAs with commodities or equity investments.