The extreme response of a jackup structure is studied. We consider how design seastate histories can be introduced to reduce the cost of time-domain response analysis. We first identify critical wave characteristics for extreme response prediction. In quasi-static cases, the maximum wave crest height, ηmax , is shown to best explain extreme deck sway. For more flexible structures we introduce a new wave characteristic, SD , based on response spectral concepts from earthquake engineering. Finally, we show how accurate response estimates can require fewer time-domain analyses, provided design seastates are pre-selected to ensure that ηmax or SD is near its average value. With respect to standard Monte-Carlo simulation, these design seastates achieve at least a 50-percent reduction in response variability, and hence at least a fourfold savings in needed simulation cost. These results may lend insight, not only into time-domain simulation, but also into more fundamental questions of jackup behavior. They also suggest that, at least in quasi-static cases, still simpler design wave methods based on ηmax may suffice. We illustrate and evaluate some such design wave methods here (e.g., the “new wave” model and others based on Slepian theory).