The paper reviews recent progress in the statistical physics of evolving networks, focusing on their structural properties in communications, biology, social sciences, and economics. It discusses the topology, evolution, and complex processes in large networks, which often exhibit scale-free properties and the "small-world" effect, where most vertices are connected by short paths. The paper explores how growing networks self-organize into scale-free structures through preferential linking, and examines topological and structural properties, percolation, and various models of network growth. Applications to real-world networks are discussed, along with connections to non-equilibrium physics, econophysics, and evolutionary biology. The paper introduces key structural characteristics of evolving networks, including degree, shortest path, clustering coefficient, and size of the giant component. It distinguishes between equilibrium and non-equilibrium networks, and discusses their properties. It covers various real-world networks, such as citation networks, collaboration networks, communications networks, the WWW, the Internet, biological networks, and electronic circuits. The paper also discusses classical random graphs, small-world networks, growing exponential networks, and scale-free networks, including the Barabási-Albert model and preferential linking. It explores percolation on networks, resilience against random breakdowns, and the role of self-organized criticality in network growth. The paper concludes with a discussion of the implications of these findings for understanding complex systems.The paper reviews recent progress in the statistical physics of evolving networks, focusing on their structural properties in communications, biology, social sciences, and economics. It discusses the topology, evolution, and complex processes in large networks, which often exhibit scale-free properties and the "small-world" effect, where most vertices are connected by short paths. The paper explores how growing networks self-organize into scale-free structures through preferential linking, and examines topological and structural properties, percolation, and various models of network growth. Applications to real-world networks are discussed, along with connections to non-equilibrium physics, econophysics, and evolutionary biology. The paper introduces key structural characteristics of evolving networks, including degree, shortest path, clustering coefficient, and size of the giant component. It distinguishes between equilibrium and non-equilibrium networks, and discusses their properties. It covers various real-world networks, such as citation networks, collaboration networks, communications networks, the WWW, the Internet, biological networks, and electronic circuits. The paper also discusses classical random graphs, small-world networks, growing exponential networks, and scale-free networks, including the Barabási-Albert model and preferential linking. It explores percolation on networks, resilience against random breakdowns, and the role of self-organized criticality in network growth. The paper concludes with a discussion of the implications of these findings for understanding complex systems.