A person's emotions and state of mind are apparent in their face and eyes. As a Latin proverb states: "The face is the portrait of the mind; the eyes, its informers.". This presents a huge challenge for computer graphics researchers in the generation of artificial entities that aim to replicate the movement and appearance of the human eye, which is so important in human-human interactions. This State of the Art Report provides an overview of the efforts made on tackling this challenging task. As with many topics in Computer Graphics, a cross-disciplinary approach is required to fully understand the workings of the eye in the transmission of information to the user. We discuss the movement of the eyeballs, eyelids, and the head from a physiological perspective and how these movements can be modelled, rendered and animated in computer graphics applications. Further, we present recent research from psychology and sociology that seeks to understand higher level behaviours, such as attention and eye-gaze, during the expression of emotion or during conversation, and how they are synthesised in Computer Graphics and Robotics.