Moving animal groups provide some of the most intriguing and difficult to characterise examples of collective behaviour. We review some recent (and not so recent) empirical research on the motion of animal groups, including fish, locusts and homing pigeons. An important concept which unifies our understanding of these groups is that of transfer of directional information. Individuals which change their direction of travel in response to the direction taken by their near neighbours can quickly transfer information about the presence of a predatory threat or food source. We show that such information transfer is optimised when the density of individuals in a group is close to that at which a phase transition occurs between random and ordered motion. Similarly, we show that even relatively small differences in information possessed by group members can lead to strong collective-level decisions for one of two options. By combining the use of self-propelled particle and social force models of collective motion with thinking about the evolution of flocking we aim to better understand how complexity arises within these groups.