The research on exception handling in Multi-Agent Systems has produced some advanced models to deal with 'exceptional situations'. The expression 'agent exception' is however unclear across the literature, as it sometimes refers to extensions of traditional exception models in programming languages, and sometimes to organizational management mechanisms with distinct semantics. In this paper, we propose a definition of 'agent exception' to clarify the notion and justify that specific research is necessary on this theme. We detail properties of this definition, revisit the traditional vocabulary related to exception in software design, propose an adequate agent architecture, and identify some research issues. This work is aimed at federating the endeavors on the question of exception management for Agent-Oriented Computing.