We briefly describe the toolkit used for studying complex systems: nonlinear dynamics, statistical physics, and network theory. We place particular emphasis on network theory--the topic of this special issue--and its importance in augmenting the framework for the quantitative study of complex systems. In order to illustrate the main issues, we briefly review several areas where network theory has led to significant developments in our understanding of complex systems. Specifically, we discuss changes, arising from network theory, in our understanding of (i) the Internet and other communication networks, (ii) the structure of natural ecosystems, (iii) the spread of diseases and information, (iv) the structure of cellular signalling networks, and (v) infrastructure robustness. Finally, we discuss how complexity requires both new tools and an augmentation of the conceptual framework--including an expanded definition of what is meant by a "quantitative prediction."