This chapter considers sliding modes applied to the problem of observer design. The historical development is outlined leading to the description of a specific class of sliding mode observer which will be used throughout the book. It will be shown how the unique properties associated with the so-called equivalent injection signal necessary to maintain sliding can be exploited to reconstruct actuator and sensor faults modelled as additive perturbations to the inputs and the outputs of the plant. Design methodologies based on Linear Matrix Inequalities (LMIs) are presented. These approaches exploit all the available degrees of freedom associated with the choice of the observer gains. The chapter describes sliding mode observers which can reconstruct faults and yet be robust to disturbances/uncertainties which may corrupt the quality of the reconstructions resulting from mismatches between the model about which the observer is designed and the real system. Initially, the design method is formulated for the case of actuator faults. A comparison is also made between the sliding mode observer schemes developed in the chapter and more traditional linear unknown input observers which are prevalent in the literature.