Sensor-fault detection is crucial for the safe operation of autonomous vehicles. This paper introduces a novel kinematics-based approach for detecting and identifying faulty sensors, which is model-independent, rule-free, and applicable to ground and aerial vehicles. This method, called kinematics-based sensor fault detection (KSFD), relies on kinematic relations, sensor measurements, and real-time single and double numerical differentiation. Using onboard data from radar, rate gyros, magnetometers, and accelerometers, KSFD uniquely identifies a single faulty sensor in real time. To achieve this, adaptive input and state estimation (AISE) is used for real-time single and double numerical differentiation of the sensor data, and the single and double transport theorems are used to evaluate the consistency of data. Unlike model-based and knowledge-based methods, KSFD relies solely on sensor signals, kinematic relations, and AISE for real-time numerical differentiation. For ground vehicles, KSFD requires six kinematics-based error metrics, whereas, for aerial vehicles, nine error metrics are used. Simulated and experimental examples are provided to evaluate the effectiveness of KSFD.