The shooter, who has not been identified, is now in custody, according to police. The suspect was tased before he was arrested, Mesa police spokesman Esteban Flores said.
Flores said there are believed to be five or six different locations where the shooting took place.
Police said six people were shot, one of whom died at the scene.
Another victim remains in critical condition; four others are expected to survive, police said.
A Mesa police spokesman said that they believe the shooting spree began inside a room at a local motel, and that the suspect then shot one person at a nearby bistro before carjacking a vehicle and driving to an apartment complex.
The suspect then broke into an apartment in that complex, shooting one man, before fleeing to a neighboring apartment building and breaking into another apartment and shooting another individual, who is now in critical condition, police said.
"We don't know exactly what happened at the first shooting," a Mesa police official said. "We know there was some type of argument and they possibly knew each other."