This paper studies the diagnosis and compensation for sensor and actuator faults in a quadrotor unmanned aerial vehicle. Without adding sensors or actuators for increased hardware redundancy, an observer-based adaptive controller is proposed to estimate and compensate for the faults. First, using a feedback linearization technique, an inner controller is designed to transform the form of the considered quadrotor unmanned aerial vehicle with faults into a nonlinear system with Lipschitz-like nonlinearities and parametric faulty models. Second, the estimations for unmeasurable state and actuator faults are performed in an output-feedback outer controller to compensate for the actuator faults. Third, a nonlinear high-gain observer is designed to provide the information of the state and faults to the outer controller, with the compensations for sensor faults. A Lyapunov-based analysis shows that appropriate choices of the controller parameters can guarantee the exponential convergence of errors in estimation and trajectory tracking under uncertainties and faults. The robustness to the external disturbances is also discussed. Simulations are given to verify the effectiveness of the proposed scheme. The proposed approach is also implemented on a quadrotor unmanned aerial vehicle to show its feasibility in real-time applications.