This paper studies, for the first time, the dependence of extreme events in energy markets. Based on a large data set comprising daily quotes of crude oil and natural gas futures, we estimate and model large comovements of commodity returns. To detect the presence of tail dependence we apply a new method based on the concept of tail copulas, which accounts for different scenarios of joint extreme outcomes. Moreover, we show that the common practice of fitting copulas to the data cannot capture the dynamics in the tail of the joint distribution. It is therefore unsuitable for risk management purposes.