This work is concerned with examining in a formal quantitative manner what human observers look at and what the objects of their gaze tell them. Three models designed to describe and predict the allocation of human attention in supervisory control tasks were investigated. A series of three experiments examined the relative influence of five factors on the sampling patterns of participants: the information generation rate of the information signal (bandwidth), the frequency of significant, i.e., task relevant, events on an information source (alarm frequency), the payoff matrix associated with missing or detecting critical events (value), the visual salience of the events, and the cost of making an observation. The paradigm employed is similar to that developed by Senders and colleagues (1964), in which observers were asked to monitor an array of four simulated ammeters and to press a button whenever the pointer of any ammeter entered an "alarm zone." Aspects of three mathematical models, Senders's constrained random sampler, Wickens and colleagues SEEV model, and Pirolli's and Card's Information Foraging Theory Model, were combined to form seven different models predicting performance in the task. The sampling patterns predicted by each model were compared against the eye movement data of participants. Results of the three experiments indicate that participants' sampling patterns were sensitive to the experimental manipulations. Comparisons of the model predicted patterns of attention allocation to those in the participant data indicated that different models described different participants. Participants who performed poorly at the task were best described by models incorporating bandwidth. Participants who performed well at the task were best described by models incorporating alarm frequency, and those who performed best at the task were not well-described by any of the models. Overall the models based on Information Foraging Theory were the most robust in predicting the attention allocation patterns of participants. Implications of each of the experimental manipulations and of the fit of the models to the participant data are discussed.