What are the laws concerning existing fences and property lines?

My mother has new neighbors. They want her to replace the existing fence between their property with "something nicer". They also claim the existing fence (which has been in place for over 40 years) is 2 feet over the property line, on their side of the property. Since the fence has been in place for as long as it has, what rights do both parties involved have, and who is responsible for the expense of removing the old fence, and constructing the new? My mom is perfectly fine with the existing fence.