Assignment Management Software??

Sorry if this question does not go with this board but I just wanted to ask if anyone had any input on this problem. Currently I have an Assignment System run in MS Access, but users want a new online version with no MS products involved. Orignally the system was going to be re-created in MySQL but I was wondering if anyone knew of any good Assignment/Project Management Software out there to make the job easier?? Free or purchase is no problem, as long as a demo is out. Please help and once again I apologize for going out of the range of this board.