  The natural world is full of complex systems characterized by intricate
relations between their components: from social interactions between
individuals in a social network to electrostatic interactions between atoms in
a protein. Topological Deep Learning (TDL) provides a comprehensive framework
to process and extract knowledge from data associated with these systems, such
as predicting the social community to which an individual belongs or predicting
whether a protein can be a reasonable target for drug development. TDL has
demonstrated theoretical and practical advantages that hold the promise of
breaking ground in the applied sciences and beyond. However, the rapid growth
of the TDL literature for relational systems has also led to a lack of
unification in notation and language across message-passing Topological Neural
Network (TNN) architectures. This presents a real obstacle for building upon
existing works and for deploying message-passing TNNs to new real-world
problems. To address this issue, we provide an accessible introduction to TDL
for relational systems, and compare the recently published message-passing TNNs
using a unified mathematical and graphical notation. Through an intuitive and
critical review of the emerging field of TDL, we extract valuable insights into
current challenges and exciting opportunities for future development.
