Networks are involved in many aspects of everyday life, from food webs in ecology and the spread of pandemics to social networking and public transportation. In fact, some of the most important and familiar natural systems and social phenomena are based on a networked structure. It is impossible to understand the spread of an epidemic, a computer virus, large-scale blackouts, or massive extinctions without taking into account the network structure that underlies all these phenomena. In this Very Short Introduction, Guido Caldarelli and Michele Catanzaro discuss the nature and variety of networks, using everyday examples from society, technology, nature, and history to illuminate the science of network theory. The authors describe the ubiquitous role of networks, reveal how networks self-organize, explain why the rich get richer, and discuss how networks can spontaneously collapse. They conclude by highlighting how the findings of complex network theory have very wide and important applications in genetics, ecology, communications, economics, and sociology.