The problem of decentralized modular fault diagnosis of concurrent discrete event systems, that is composed of a set of component modules, is formulated and studied. In the proposed decentralized modular framework, diagnosis is performed by the local diagnosers, located at the component sites, using their own local observations. This is to ensure the scalability of the approach with respect to the number of component modules, and we require that the local diagnosers be "modularly computable", i.e., their computation should be based on the local models, and not the global models. It is also required that there are no missed-detections (every fault is detected within a bounded number of transitions) and no false-alarms (a fault detection report is issued only when a fault has occurred). We formally define the decentralized modular diagnosis problem and introduce the notion of modular diagnosability as a key property for the existence of desired decentralized modular diagnosers. We show that under this property, the complexity for constructing the local diagnosers is polynomial in the number of local modules. We present a method for testing the modular diagnosability property by reducing it to an instance of a certain codiagnosability property for which known verification techniques exist.