Neighbor's tree branches hang over my driveway. Who is responsible for trimming these branches?

If your neighbor's tree branches are hanging onto your property they are not legally obligated to trim them.  However, you have the right to do so but at your expense.  However, if the branches are interfering with any utility wires or the like, then contact the utility company; they can make your neighbor trim them.  Additionally, if they are creating any type of safety issue contact you local zoning department, they can force your neighbor to remove any obstruction in such a case.

You may also want to discuss this with your neighbor (if you have not already done so).  You can let them know that any damage that is done to your property, if for example, one of the branches were to fall and hit your car, would be their responsibility and they will legally be liable to you.  Even if they won't pay for the full cost of the tree trimming possibly they will share the cost with you.