In order to more effectively cope with the real-world problems of vagueness, fuzzy discrete-event systems (FDESs) were proposed by Lin and Ying recently. Then we and Cao and Ying investigated the supervisory control of FDESs independently. In this paper, we are concerned with another important issue of FDESs, the failure diagnosis. More specifically: (1) we propose a ldquofuzzy diagnosabilityrdquo approach by introducing a fuzzy diagnosability function to characterize the diagnosability degree, which takes values in the interval [0,1] rather than { 0,1}; (2) based on the observability of events, we formalize the construction of the diagnosers that are used to perform fuzzy diagnosis; (3) a number of basic properties of the diagnosers are investigated. In particular, we present a necessary and sufficient condition for failure diagnosis of FDESs. Our results generalize the important consequences of the diagnosability for crisp discrete-event systems (DESs) introduced by Sampath et al. The newly proposed approach allows us to deal with the problem of diagnosability for both crisp DESs and FDESs; (4) in addition, a method for checking the fuzzy diagnosability for FDESs is proposed. Also, some examples are provided to illustrate the application of the diagnosability of FDESs.