A gas concentration measuring apparatus for use in air-fuel ratio control of motor vehicle engines is provided which is designed to detect and identify a cause of failure in operation of an A/F sensor. The apparatus works to apply the voltage to a sensor element and sweep it to produce a change in electrical current flowing through the sensor element for measuring the impedance of the sensor element. The apparatus samples values of the change in the current, a sensor output indicating an air-fuel ratio of mixture supplied to the engine, etc., before and/or after the sensor element is activated and uses a combination thereof to identify the cause of the failure of the A/F sensor.