In this seminar we discuss how by considering eye movements, and in particular the resulting sequence of gaze shifts, a stochastic process, a wide variety of tools become available for analyses and modelling beyond conventional statistical methods. We first give a brief, though critical, probabilistic tour of current computational models of eye movements and visual attention, and we lay down the basis for gaze shift pattern analysis. Then we discuss their links to the concepts of Markov Processes, the Wiener process and related random walks within the Gaussian framework of the Central Limit Theorem Eventually we will deliberately violate the fundamental assumptions of the Central Limit Theorem to elicit a larger perspective, rooted in statistical physics, for analysing and modelling eye movements in terms of anomalous, non-Gaussian, random walks and modern foraging theory.