Fault detection and isolation is a crucial and challenging task in the automatic control of large complex systems. We propose a discrete-event system (DES) approach to the problem of failure diagnosis, We introduce two related notions of diagnosability of DES’s in the framework of formal languages and compare diagnosability with the related notions of observability and invertibility. We present a systematic procedure for detection and isolation of failure events using diagnosers and provide necessary and sufficient conditions for a language to be diagnosable, The diagnoser performs diagnostics using on-line observations of the system behavior; it is also used to state and verify off-line the necessary and sufficient conditions for diagnosability, These conditions are stated on the diagnoser or variations thereof. The approach to failure diagnosis presented in this paper is applicable to systems that fall naturally in the class of DES’s; moreover, for the purpose of diagnosis, most continuous variable dynamic systems can be viewed as DES’s at a higher level of abstraction, In a companion paper [20], we provide a methodology for building DES models for the purpose of failure diagnosis and present applications of the theory developed in this paper.