  Real networks are finite metric spaces. Yet the geometry induced by shortest
path distances in a network is definitely not its only geometry. Other forms of
network geometry are the geometry of latent spaces underlying many networks,
and the effective geometry induced by dynamical processes in networks. These
three approaches to network geometry are all intimately related, and all three
of them have been found to be exceptionally efficient in discovering
fractality, scale-invariance, self-similarity, and other forms of fundamental
symmetries in networks. Network geometry is also of great utility in a variety
of practical applications, ranging from the understanding how the brain works,
to routing in the Internet. Here, we review the most important theoretical and
practical developments dealing with these approaches to network geometry in the
last two decades, and offer perspectives on future research directions and
challenges in this novel frontier in the study of complexity.
