We review a recent fast progress in statistical physics of evolving networks. Interest focuses mainly on the structure properties of random hierarchically organized networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind were created recently. This opens a wide field for research of their topology, evolution, and complex processes proceeding in them. Such networks possess a rich set of scaling properties. A number of them is scale-free and show striking resilience against random breakdowns. In spite of huge sizes of these networks, the distances between most of nodes of the networks are short -- the ``small-world'' effect. Their features make them appropriate for numerous applications. We discuss how growing networks self-organize into scale-free structure and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in networks. We present a number of models demonstrating the main features of evolving networks and discuss existing approaches to their simulation and analytical study. Applications of the general results to the particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc.