  Environmental data science for spatial extremes has traditionally relied
heavily on max-stable processes. Even though the popularity of these models has
perhaps peaked with statisticians, they are still perceived and considered as
the `state-of-the-art' in many applied fields. However, while the asymptotic
theory supporting the use of max-stable processes is mathematically rigorous
and comprehensive, we think that it has also been overused, if not misused, in
environmental applications, to the detriment of more purposeful and
meticulously validated models. In this paper, we review the main limitations of
max-stable process models, and strongly argue against their systematic use in
environmental studies. Alternative solutions based on more flexible frameworks
using the exceedances of variables above appropriately chosen high thresholds
are discussed, and an outlook on future research is given, highlighting
recommendations moving forward and the opportunities offered by hybridizing
machine learning with extreme-value statistics.
