Reasoning about intentional action is a pervasive and critical skill in the human cognitive repertoire. Intentions have taken center-stage in discussions of how humans parse perceptual input, understand language, make moral judgments, and predict the behavior of conspecifics. In the quest to engineer machine intelligence, intentions have largely either been ignored entirely, or have been given oversimplified construals as either preference-orders over actions, or as simple predicates in computational theories of action. In this paper, we will motivate the need for intelligent systems capable of reasoning about the intentions of others by presenting a number of germane application areas, including those which deal with the integration of intention recognition with other cognitive processes including dialogue processing. We then briefly review the relevant psychological literature on the development and operation of the human capacity to recognize and reason about the intentions of others. In doing so, we will extract a number of desiderata for the development of computational models of intention-recognition. We will then show how these requirements motivate principled design choices for the construction of intelligent systems. Finally, we will close with a brief description of Polyscheme, a computational cognitive architecture which we feel sufficiently addresses the computational challenges entailed by our desiderata.