Inertial sensors relying on atom interferometry offer a breakthrough advance in a variety of applications, such as inertial navigation, gravimetry or ground- and space-based tests of fundamental physics. These instruments require a quiet environment to reach their performance and using them outside the laboratory remains a challenge. Here we report the first operation of an airborne matter-wave accelerometer set up aboard a 0g plane and operating during the standard gravity (1g) and microgravity (0g) phases of the flight. At 1g, the sensor can detect inertial effects more than 300 times weaker than the typical acceleration fluctuations of the aircraft. We describe the improvement of the interferometer sensitivity in 0g, which reaches 2 x 10-4 ms-2 / √Hz with our current setup. We finally discuss the extension of our method to airborne and spaceborne tests of the Universality of free fall with matter waves.