This survey reviews the theory and applications of complex oscillator networks, focusing on phase oscillator models that generalize the Kuramoto model. These models are widely used in synchronization phenomena and exhibit rich dynamic behavior. The paper discusses the importance of coupled oscillator models as locally canonical models and their applications in control science, including vehicle coordination, electric power networks, and clock synchronization. It introduces synchronization notions such as frequency synchronization, phase synchronization, phase balancing, and pattern formation. The paper presents the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with various interconnection topologies and in both finite- and infinite-dimensional settings. It also highlights the limitations of existing analysis methods and suggests directions for future research. The paper reviews the history of synchronization studies, the development of the Kuramoto model, and its applications in various scientific and engineering fields. It discusses the mechanical analog of the coupled oscillator model and its relevance to synchronization phenomena. The paper also presents selected applications in sciences and engineering, including biological synchronization, electric power networks, and clock synchronization in decentralized networks. It introduces the canonical coupled oscillator model and its derivation from weakly coupled limit-cycle oscillators. The paper discusses synchronization notions, metrics, and analysis methods for both finite and infinite-dimensional systems. It concludes by summarizing the limitations of existing analysis methods and highlighting important directions for future research.This survey reviews the theory and applications of complex oscillator networks, focusing on phase oscillator models that generalize the Kuramoto model. These models are widely used in synchronization phenomena and exhibit rich dynamic behavior. The paper discusses the importance of coupled oscillator models as locally canonical models and their applications in control science, including vehicle coordination, electric power networks, and clock synchronization. It introduces synchronization notions such as frequency synchronization, phase synchronization, phase balancing, and pattern formation. The paper presents the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with various interconnection topologies and in both finite- and infinite-dimensional settings. It also highlights the limitations of existing analysis methods and suggests directions for future research. The paper reviews the history of synchronization studies, the development of the Kuramoto model, and its applications in various scientific and engineering fields. It discusses the mechanical analog of the coupled oscillator model and its relevance to synchronization phenomena. The paper also presents selected applications in sciences and engineering, including biological synchronization, electric power networks, and clock synchronization in decentralized networks. It introduces the canonical coupled oscillator model and its derivation from weakly coupled limit-cycle oscillators. The paper discusses synchronization notions, metrics, and analysis methods for both finite and infinite-dimensional systems. It concludes by summarizing the limitations of existing analysis methods and highlighting important directions for future research.