In this paper, fault detection and diagnosis (FDD) of faults with independent effect on system outputs by using the adaptive observer technique are investigated. At first, a class of linear systems without model uncertainty is considered. Then, a general situation where the system is subjected to either model errors or external disturbance is discussed. Robust adaptive control techniques are applied to guarantee convergence of certain signals to residual sets. An extension to FDD for a class of non-linear systems with non-linear fault function is extensively investigated. The novelty of this paper is that the strict positive realness (SPR) requirement on the plant transfer function in existing results is removed at the expense of requiring the existence of a positive definite solution to a certain matrix inequality. Furthermore, the problems of stabilization and robust stabilization by fault-tolerant control (FTC) and robust FTC are studied respectively, and fault-tolerant controllers are designed to stabilize the close-loop systems. An aircraft example and a numerical example are included to verify the applicability of the proposed diagnosis methods.