Exceptional cases are at odds with the typical: they stand out as bizarre and rare. What then could justify their systematic analysis? Elaborating the analytical distinction between anomalies, exceptions and outliers, this paper outlines three potential epistemic contributions of exceptional cases. First, exceptional cases reveal the limits of standard classification categories. In so doing, they problematize usual classificatory grids. Their input is critical. Second, exceptional cases point to new classes of objects. They acquire paradigmatic status when they exemplify the characteristic features of these new classes with utmost clarity. Third, exceptional cases magnify relational patterns that in more mundane contexts lack visibility. Here their contribution is heuristic. These three contributions become possible when we put at bay normative expectations of what should happen, and specify cases by reference to an analytical space of constitutive dimensions. To underscore the general significance of these observations, I draw on examples borrowed from different quarters of the social sciences: the sociology of organizations, ethnomethodology, comparative historical sociology and the history of science.