The Internet and the Web can be described as huge networks of connected computers, connected Web pages or connected users. The formal structure of these networks is expected to represent patterns of communication and organization, and to influence the nature of communication in these networks. A number of approaches have been developed to study these phenomena. This paper reviews the emergence from theoretical physics of a new specialty in complexity theory which analyses the Internet and the Web as complex networks. Concepts and findings from this area of physics are reviewed and made accessible to a non-physics audience. In complexity theory, the concept of connectivity is expressed by mathematical laws, addressing the distribution of links over nodes, the emergence of hierarchies or the behavior of “inhabitants” of such networks. The paper begins with an introduction to the topological classification of complex networks as “small world networks” and “scale-free networks.” It discusses how specific topologies or connectivity patterns are based on the construction and growth of such networks. Major findings about the Internet and the Web are discussed. The paper also explores the possibilities of linking statistical empirical analysis and mathematical modeling to qualitative research as a way of gaining insight into the emergence of complex networks.