The suspect ran off, but police caught up to him and arrested him. A gun was recovered from the scene.
On Sunday morning, Dallas police identified the suspect as 22-year-old Cameron Turner.
Turner is charged with Aggravated Assault.
Dallas police say that the victims in the shooting are two adult males and one adult female.
The State Fair of Texas X account posted about a State Fair Emergency Order and that the fairgrounds were being evacuated.