Critical phenomena in complex networks involve unique behaviors arising from the compactness and complex structures of networks, differing from traditional cooperative systems on lattices. This review explores structural phase transitions and critical phenomena in complex networks, including percolation, epidemic thresholds, synchronization, and self-organized criticality. It covers equilibrium and growing networks, discussing the birth of the giant connected component, condensation transitions, and critical effects in spin models, epidemic spreading, and other models like the Ising, Potts, and XY models. The paper also addresses finite size effects, open problems, and future directions. Key topics include the role of clustering, small-world networks, and the impact of network architecture on critical phenomena. The review emphasizes the importance of analytical methods such as the Bethe ansatz, belief propagation, and statistical mechanics in understanding these phenomena. It highlights the differences between equilibrium and growing networks, the significance of degree distributions, and the influence of network topology on critical behavior. The paper concludes with a discussion of open problems and the potential for further research in this rapidly evolving field.Critical phenomena in complex networks involve unique behaviors arising from the compactness and complex structures of networks, differing from traditional cooperative systems on lattices. This review explores structural phase transitions and critical phenomena in complex networks, including percolation, epidemic thresholds, synchronization, and self-organized criticality. It covers equilibrium and growing networks, discussing the birth of the giant connected component, condensation transitions, and critical effects in spin models, epidemic spreading, and other models like the Ising, Potts, and XY models. The paper also addresses finite size effects, open problems, and future directions. Key topics include the role of clustering, small-world networks, and the impact of network architecture on critical phenomena. The review emphasizes the importance of analytical methods such as the Bethe ansatz, belief propagation, and statistical mechanics in understanding these phenomena. It highlights the differences between equilibrium and growing networks, the significance of degree distributions, and the influence of network topology on critical behavior. The paper concludes with a discussion of open problems and the potential for further research in this rapidly evolving field.