I'm not an electrician but I believe that the NEC national electrical code does require the Romex wire to be stapled to the wall studs to pass inspection.
That is if you're getting your electrical work inspected. I have just run it through some of my walls without stapling it since I didn't require an inspection.
Whenever the wire passes through a stud you should put a metal plate that will stop nails or screws from going through the wire in the event someone drives a nail or screw through the stud.
The metal plate will protect the wire and the person from getting shocked by stopping the nail or screw from going through the stud.