  Modeling complex systems and data using the language of graphs and networks
has become an essential topic across a range of different disciplines.
Arguably, this network-based perspective derives is success from the relative
simplicity of graphs: A graph consists of nothing more than a set of vertices
and a set of edges, describing relationships between pairs of such vertices.
This simple combinatorial structure makes graphs interpretable and flexible
modeling tools. The simplicity of graphs as system models, however, has been
scrutinized in the literature recently. Specifically, it has been argued from a
variety of different angles that there is a need for higher-order networks,
which go beyond the paradigm of modeling pairwise relationships, as
encapsulated by graphs. In this survey article we take stock of these recent
developments. Our goals are to clarify (i) what higher-order networks are, (ii)
why these are interesting objects of study, and (iii) how they can be used in
applications.
