The single-engine Consolidated Vultee BT-13A ran off the runway and came to rest along a perimeter fence around 3:35 p.m. while the pilot was practicing touch-and-goes at Livermore Municipal Airport
A touch-and-go is a procedure where the pilot lands and then takes off again without stopping.
Neither of the two people on board was injured.
The plane landed about 630 feet from the end of the runway near the sixth hole of the Las Positas Golf Course.
No further details were immediately available on the cause of the crash. The FAA is investigating.