This review article focuses on the synchronization of oscillating elements in complex networks, a topic that has gained significant attention due to its relevance in various fields such as biology, engineering, and social sciences. The authors provide an overview of the recent advances in understanding synchronization phenomena when oscillating elements are constrained to interact in complex network topologies. They discuss the interplay between the structural characteristics of the network and the functional behavior of the oscillators, highlighting the emergence of new features that arise from this interplay. The article begins with an introduction to complex networks, covering key concepts such as degree distribution, average shortest path length, and clustering coefficient. It then delves into the synchronization of coupled phase oscillators, including the Kuramoto model and its extensions to complex networks. The authors explore the onset of synchronization in complex networks, presenting both numerical and analytical approaches to determine the critical coupling strength. They also discuss the stability of the synchronized state using the Master Stability Function (MSF) formalism and the design of synchronizable networks. The review further examines the applications of synchronization in complex networks across various disciplines, including biological systems, neuroscience, engineering, computer science, and social sciences. Finally, the authors provide perspectives and conclusions, emphasizing the importance of understanding the interplay between network topology and synchronization dynamics in complex systems.This review article focuses on the synchronization of oscillating elements in complex networks, a topic that has gained significant attention due to its relevance in various fields such as biology, engineering, and social sciences. The authors provide an overview of the recent advances in understanding synchronization phenomena when oscillating elements are constrained to interact in complex network topologies. They discuss the interplay between the structural characteristics of the network and the functional behavior of the oscillators, highlighting the emergence of new features that arise from this interplay. The article begins with an introduction to complex networks, covering key concepts such as degree distribution, average shortest path length, and clustering coefficient. It then delves into the synchronization of coupled phase oscillators, including the Kuramoto model and its extensions to complex networks. The authors explore the onset of synchronization in complex networks, presenting both numerical and analytical approaches to determine the critical coupling strength. They also discuss the stability of the synchronized state using the Master Stability Function (MSF) formalism and the design of synchronizable networks. The review further examines the applications of synchronization in complex networks across various disciplines, including biological systems, neuroscience, engineering, computer science, and social sciences. Finally, the authors provide perspectives and conclusions, emphasizing the importance of understanding the interplay between network topology and synchronization dynamics in complex systems.