Artistic Gymnastics:
-For Women : Uneven Bars, Floor Balance Beam, Vault
-For Men : Floor, Pommel Horse, Still Rings, Vault, Parallel Bars, High Bar
Rhythmic Gymnastics (usually women): ribbon, clubs, rope, hoop, ball and sometimes floor routine (called Free (routine)).
Trampolining and Tumbling: individual, synchronized, double mini and power tumbling.
Acrobatic Gymnastics (aka Sports Acrobatics) (you usually do it in school, at least in France you do)
Team Gym.
Display Gymnastics
Aerobic Gymnastics.
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Women's Artistic gymnastics: Floor, Beam, Bars, Vault.
Men's Artistic gymnastics: Pommel Horse, High Bar, Parallel Bars, Floor, Rings.
Trampoline and Tumbling: Trampoline and Tumbling.
Rhythmic gymnastics: Clubs, Ribbon, Ball, Hoop.
Acro: They do routines together not quiet sure how.
Those are at least most of them i hope it helped