  Modeling of phenomena such as anomalous transport via fractional-order
differential equations has been established as an effective alternative to
partial differential equations, due to the inherent ability to describe
large-scale behavior with greater efficiency than fully-resolved classical
models. In this review article, we first provide a broad overview of
fractional-order derivatives with a clear emphasis on the stochastic processes
that underlie their use. We then survey three exemplary application areas -
subsurface transport, turbulence, and anomalous materials - in which
fractional-order differential equations provide accurate and predictive models.
For each area, we report on the evidence of anomalous behavior that justifies
the use of fractional-order models, and survey both foundational models as well
as more expressive state-of-the-art models. We also propose avenues for future
research, including more advanced and physically sound models, as well as tools
for calibration and discovery of fractional-order models.
