  Understanding the dissemination of diseases, information, and behavior stands
as a paramount research challenge in contemporary network and complex systems
science. The COVID-19 pandemic and the proliferation of misinformation are
relevant examples of the importance of these dynamic processes, which have
recently gained more attention due to the potential of higher-order networks to
unlock new avenues for their investigation. Despite being in its early stages,
the examination of social contagion in higher-order networks has witnessed a
surge of novel research and concepts, revealing different functional forms for
the spreading dynamics and offering novel insights. This review presents a
focused overview of this body of literature and proposes a unified formalism
that covers most of these forms. The goal is to underscore the similarities and
distinctions among various models, to motivate further research on the general
and universal properties of such models. We also highlight that while the path
for additional theoretical exploration appears clear, the empirical validation
of these models through data or experiments remains scant, with an unsettled
roadmap as of today. We therefore conclude with some perspectives aimed at
providing possible research directions that could contribute to a better
understanding of this class of dynamical processes, both from a theoretical and
a data-oriented point of view.
