The paper deals with the challenge of predicting the extreme response of catenary risers, a topic of both industry and academic interest. Large heave motions introduced at the upper end of a catenary riser can lead to compression and large bending moments in the region immediately above the touch down area. In the worst case, dynamic beam buckling may occur. The focus of the paper will be on understanding the riser behaviour in extreme, low-tension response and in establishing suitable analysis strategies to predict the extreme response. Results from long nonlinear stochastic simulations of many sea states with varying environmental and operating conditions may be combined to describe the long-term response of a nonlinear structure such as a catenary riser. However, this theoretically straight-forward approach is very demanding computationally and ways to limit the extent of nonlinear stochastic simulations are therefore sought. The usefulness of simpler methods such as regular wave analysis to improve understanding of the physical behaviour and to aid in concentrating the nonlinear simulations to where they are most useful, will be demonstrated.