  Synchronization of an ensemble of oscillators is an emergent phenomenon
present in several complex systems, ranging from social and physical to
biological and technological systems. The most successful approach to describe
how coherent behavior emerges in these complex systems is given by the
paradigmatic Kuramoto model. This model has been traditionally studied in
complete graphs. However, besides being intrinsically dynamical, complex
systems present very heterogeneous structure, which can be represented as
complex networks. This report is dedicated to review main contributions in the
field of synchronization in networks of Kuramoto oscillators. In particular, we
provide an overview of the impact of network patterns on the local and global
dynamics of coupled phase oscillators. We cover many relevant topics, which
encompass a description of the most used analytical approaches and the analysis
of several numerical results. Furthermore, we discuss recent developments on
variations of the Kuramoto model in networks, including the presence of noise
and inertia. The rich potential for applications is discussed for special
fields in engineering, neuroscience, physics and Earth science. Finally, we
conclude by discussing problems that remain open after the last decade of
intensive research on the Kuramoto model and point out some promising
directions for future research.
