The paper reviews the critical phenomena in complex networks, which are characterized by compactness and complex architectures. These features lead to a wide range of non-traditional critical effects, including structural phase transitions and transitions in cooperative models on networks. The authors discuss two main classes of critical phenomena: structural phase transitions in network architectures and transitions in cooperative models on networks. They also explore systems where networks and interacting agents influence each other. The review covers a broad spectrum of critical phenomena, such as the birth of the giant connected component, percolation, $k$-core percolation, epidemic thresholds, condensation transitions, critical phenomena in spin models on networks, synchronization, and self-organized criticality. The paper emphasizes the strong finite-size effects in these systems and highlights open problems and future directions.The paper reviews the critical phenomena in complex networks, which are characterized by compactness and complex architectures. These features lead to a wide range of non-traditional critical effects, including structural phase transitions and transitions in cooperative models on networks. The authors discuss two main classes of critical phenomena: structural phase transitions in network architectures and transitions in cooperative models on networks. They also explore systems where networks and interacting agents influence each other. The review covers a broad spectrum of critical phenomena, such as the birth of the giant connected component, percolation, $k$-core percolation, epidemic thresholds, condensation transitions, critical phenomena in spin models on networks, synchronization, and self-organized criticality. The paper emphasizes the strong finite-size effects in these systems and highlights open problems and future directions.