A design of reduced-order observers is proposed for linear time-invariant systems with unknown inputs including unknown measurement disturbances. Based on the unknown-input observer design, a systematic treatment of designing fault detection and isolation (FDI) observers is provided. The core of the FDI observer design is to find an observable subsystem where the faults of interest show up and the rest of the faults disappear. The FDI observer design procedure has a clear physical meaning. Robust FDI observers for uncertain systems can also be obtained using the FDI observer design provided the uncertainties of the system can be formulated as unknown inputs. It is shown that the parity equations of another important FDI method can be generated in an elegant way using a dead-beat FDI observer scheme directly in the time domain. Examples are included to show some concepts and to illustrate the FDI observer design method.