This survey reviews the theory and applications of complex oscillator networks, focusing on phase oscillator models that generalize the Kuramoto model. The authors discuss the history and applications of these models in various scientific and engineering fields, including biology, physics, chemistry, social networks, and technological applications such as vehicle coordination, electric power networks, and clock synchronization. They introduce key synchronization notions, including frequency and phase synchronization, phase balancing, pattern formation, and partial synchronization, and present analysis approaches for these concepts. The survey covers the latest results on synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. It concludes by highlighting limitations of existing analysis methods and suggesting future research directions.This survey reviews the theory and applications of complex oscillator networks, focusing on phase oscillator models that generalize the Kuramoto model. The authors discuss the history and applications of these models in various scientific and engineering fields, including biology, physics, chemistry, social networks, and technological applications such as vehicle coordination, electric power networks, and clock synchronization. They introduce key synchronization notions, including frequency and phase synchronization, phase balancing, pattern formation, and partial synchronization, and present analysis approaches for these concepts. The survey covers the latest results on synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. It concludes by highlighting limitations of existing analysis methods and suggesting future research directions.