Vaulting is one of the seven equestrian disciplines recognized by the Fédération Equestre Internationale (FEI) for international competition
It is most easily described as gymnastics performed on the back (or other parts) of a moving horse.
Participants are judged on both a set of compulsory moves, and in freestyle competition called (in German) a Kür.
The USA Equestrian Rule Book defines Vaulting:
Vaulting is the art of gymnastics on the moving horse. The competition is judged on the smooth and correct execution of the compulsory exercises and free-style programs by the vaulters in sympathy and harmony with the horse working on the longe line.