From the Super Mario Wiki
Owls are large helpful birds that appear in Wario Land II and Wario Land 3. They are usually seen sleeping, but awake if Wario touches them. Then they will carry Wario around if he sticks to their feet. The player can freely change the flight direction by pressing . If Wario touches a dangerous obstacle or gets hit by an enemy during the flight, the owl will escape and return to the place where Wario found it. In Wario Land 3, owls have their own music for when Wario grabs onto them.
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