  The classical modeling of spatial extremes relies on asymptotic models (i.e.,
max-stable processes or $r$-Pareto processes) for block maxima or peaks over
high thresholds, respectively. However, at finite levels, empirical evidence
often suggests that such asymptotic models are too rigidly constrained, and
that they do not adequately capture the frequent situation where more severe
events tend to be spatially more localized. In other words, these asymptotic
models have a strong tail dependence that persists at increasingly high levels,
while data usually suggest that it should weaken instead. Another well-known
limitation of classical spatial extremes models is that they are either
computationally prohibitive to fit in high dimensions, or they need to be
fitted using less efficient techniques. In this review paper, we describe
recent progress in the modeling and inference for spatial extremes, focusing on
new models that have more flexible tail structures that can bridge asymptotic
dependence classes, and that are more easily amenable to likelihood-based
inference for large datasets. In particular, we discuss various types of random
scale constructions, as well as the conditional spatial extremes model, which
have recently been getting increasing attention within the statistics of
extremes community. We illustrate some of these new spatial models on two
different environmental applications.
