In my uni, the policies differ from department to department, and also the nature of the assignment that was given. If it's on a normal homework assignment, usually it's zero credit for that assignment, with subsequent offences leading to an automatic failing grade, and probably some comments inserted into your school record.

However, if it's something like a term paper, those are dealt a lot more severely, with an automatic failing grade, and the possibility of being dropped from the university as well.