Several existing detectability and isolability definitions are reviewed. It is argued that two types of definitions have to be distin guished. On onehand, intrinsic definitions capture the signature of the fault on the system. On the other hand, performance-based definitions involve indexes of performance of fault detection algorithms. In particular, since many fault detection algorithms involvearesidual generation mechanism, such definitions may capture, among other performance indexes, the signature of the fault on the residuals. In the particular case of sensor and actuator faults in linear dynamic systems, an equivalence among two defin itio ns is exhib ite d. T he usefulne ss of some definitions for component faults in nonlinear systems is illustrated inasimulated example.