Much attention has been focused recently on the topic of complex network behavior [1-5]. Most of the interest has been on scale-free networks, which are believed to represent many phenomena in the nature. Scale-free degree distributions have been observed in the Internet[6],World Wide Web (WWW) [7], metabolic networks [8] and many others. For recent reviews see [9-13]. In this paper we review the topics of structure [14] and percolation of such networks [1-4]. Understanding network structure can help devise better network topologies. It may also help design more efficient algorithms for routing and searching in communication networks by taking advantage of the network structure. Percolation is especially important in forecasting and preventing network malfunctions in the Internet, as well as other realistic networks, and may also be important in the understanding of the stability of biological and chemical processes [15].