Observer-based fault diagnosis methods, thanks to their intimate relations to automatic control, are widely accepted as a powerful technique in dealing with process monitoring and fault diagnosis in dynamic processes. In the unified framework of control and detection, it is natural to make use of the available detection capability in the observer-based feedback controller realisation to achieve fault detection and estimation. In this chapter, we firstly review some basic observer-based methods for residual generation, which can be applied in fact both for control and fault detection purposes.