Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many realworld networks interact with and depend on each other. Very recently an analytical D. Y. Kenett (B) · J. Gao · X. Huang · S. Shao · G. Paul · H. E. Stanley Center for Polymer Studies, Department of Physics, Boston university, Boston, MA 02215, USA e-mail: email@example.com X. Huang e-mail: firstname.lastname@example.org S. Shao e-mail: email@example.com G. Paul e-mail: firstname.lastname@example.org H. E. Stanley e-mail: email@example.com J. Gao Department of Automation, Shanghai Jiao Tong, University, 800 Dongchuan Road, Shanghai 200240, People’s Republic of China Center for Complex Network Research and Department of Physics, Northeastern University, Boston, MA02115, USA e-mail: firstname.lastname@example.org I. Vodenska Administrative Sciences Department, Metropolitan College, Boston University, Boston, MA 02215, USA e-mail: email@example.com S. V. Buldyrev Department of Physics, Yeshiva University, New York, NY10033, USA e-mail: firstname.lastname@example.org S. Havlin Department of Physics, Bar-Ilan University, Ramat Gan, Israel e-mail: email@example.com G. D’Agostino and A. Scala (eds.), Networks of Networks: The Last Frontier of Complexity, 3 Understanding Complex Systems, DOI: 10.1007/978-3-319-03518-5_1, © Springer International Publishing Switzerland 2014 4 D. Y. Kenett et al. framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a network of networks (NON) formed by n interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, although networks with broad degree distributions, e.g., scale-free networks, are robust when analyzed as single networks, they become vulnerable in a NON. Moreover, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (is a first-order transition), unlike the well-known continuous second-order transition in single isolated networks. We also review some possible real-world applications of NON theory.