Many animals live in societies where individuals frequently interact socially with each other. The social structures of these systems can be studied in depth by means of network analysis. A large number of studies on animal social networks in many species have in recent years been carried out in the biological research field of animal behaviour and have provided new insights into behaviour, ecology, and social evolution. This line of research is currently not so well connected to the field of complex systems as could be expected. The purpose of this paper is to provide an introduction to animal social networks for complex systems scientists and highlight areas of synergy. We believe that an increased integration of animal social networks with the interdisciplinary field of complex systems and networks would be beneficial for various reasons. Increased collaboration between researchers in this field and biologists studying animal social systems could be valuable in solving challenges that are of importance to animal social network research. Furthermore, animal social networks provide the opportunity to investigate hypotheses about complex systems across a range of natural real-world social systems. In this paper, we describe what animal social networks are and main research themes where they are studied; we give an overview of the methods commonly used to study animal social networks; we highlight challenges in the study of animal social networks where complex systems expertise may be particularly valuable; and we consider aspects of animal social networks that may be of particular interest to complex systems researchers. We hope that this will help to facilitate further interdisciplinary collaborations involving animal social networks, and further integration of these networks into the field of complex systems.