Synchronization in complex networks is a key phenomenon in many natural and artificial systems, including biological, ecological, technological, and social systems. This review discusses the advances in understanding synchronization in populations of oscillating elements constrained to interact on complex network topologies. It explores how the structure of the network influences synchronization, and presents both numerical and analytical approaches to the problem. The review also covers various applications of synchronization in complex networks across different disciplines, including biological systems, engineering, and social sciences. The review begins with an introduction to synchronization as an emerging phenomenon in dynamically interacting units. It then provides an overview of complex networks, their structural characteristics, and key measures such as degree distribution, average shortest path length, and clustering coefficient. The review then focuses on synchronization of populations of oscillators, particularly phase oscillators, and presents the Kuramoto model as a fundamental framework for understanding synchronization in complex networks. The review discusses the stability of the synchronized state in complex networks using the Master Stability Function (MSF) formalism. It also explores the design of synchronizable networks and the role of network topology in synchronization. The review then presents various applications of synchronization in complex networks, including biological systems, computer science, and social sciences. Finally, the review concludes with a discussion of the future directions and challenges in the study of synchronization in complex networks. Key findings include the role of network topology in determining the onset of synchronization, the influence of network structure on synchronization dynamics, and the importance of network heterogeneity in synchronization processes. The review also highlights the challenges in analyzing synchronization in complex networks, particularly in the thermodynamic limit, and the need for further theoretical and numerical studies to better understand the synchronization phenomena in complex systems.Synchronization in complex networks is a key phenomenon in many natural and artificial systems, including biological, ecological, technological, and social systems. This review discusses the advances in understanding synchronization in populations of oscillating elements constrained to interact on complex network topologies. It explores how the structure of the network influences synchronization, and presents both numerical and analytical approaches to the problem. The review also covers various applications of synchronization in complex networks across different disciplines, including biological systems, engineering, and social sciences. The review begins with an introduction to synchronization as an emerging phenomenon in dynamically interacting units. It then provides an overview of complex networks, their structural characteristics, and key measures such as degree distribution, average shortest path length, and clustering coefficient. The review then focuses on synchronization of populations of oscillators, particularly phase oscillators, and presents the Kuramoto model as a fundamental framework for understanding synchronization in complex networks. The review discusses the stability of the synchronized state in complex networks using the Master Stability Function (MSF) formalism. It also explores the design of synchronizable networks and the role of network topology in synchronization. The review then presents various applications of synchronization in complex networks, including biological systems, computer science, and social sciences. Finally, the review concludes with a discussion of the future directions and challenges in the study of synchronization in complex networks. Key findings include the role of network topology in determining the onset of synchronization, the influence of network structure on synchronization dynamics, and the importance of network heterogeneity in synchronization processes. The review also highlights the challenges in analyzing synchronization in complex networks, particularly in the thermodynamic limit, and the need for further theoretical and numerical studies to better understand the synchronization phenomena in complex systems.