Extreme value analysis studies rare events beyond the range of observed data. This requires models that are fitted to available data and justified by asymptotic theory. In this talk, I give an overview of two extrapolation approaches in extreme value theory, based on peaks-over-threshold methods and max-stable limits, and place my research in this context. In the first part, I present recent results on central limit theory for peaks-over-threshold estimators applied to long-memory linear time series. These simple long-memory models can exhibit clustering of extremes and are therefore challenging to study in an extreme value context. In the second part, I consider max-stable processes, which can be viewed as pointwise maxima over collections of underlying events, such as storm trajectories. While max-stable models are asymptotically justified, they can be too rigid to fit real-world data. I construct stationary processes that are linked to max-stable processes through their so-called domains of attraction. This is achieved by introducing time changes of the underlying events, leading to a broad class of stationary models with the same limiting behavior but a more flexible structure. The talk presents two directions in my research on extremes under dependence.