@Ola Olson,
It depends. Ragman is correct that changing a name is a component of a United States divorce, and one may elect to change one's name during that process. If one is not getting a divorce (or if one is already divorced and did not change one's name), one can apply for a name change in a civil proceeding, not necessarily at "City Hall" but perhaps at the County Courthouse, depending on where you live, of course. Then, once legally changed, one goes to the Social Security Administration and Drivers License bureau, to process the appropriate forms.
I have no experience with how one changes one's name in Norway. But if one were to check google one might find that all one needs to do is notify one's local tax office so one's new name may be registered correctly in the Norwegian National Registry (Folkeregisteret). Likewise, if one's name has ever been changed it is one's responsibility to make sure that the name change has been registered in the National Registry. Google is your friend.