The A320 series will do the whole thing fully managed. It essentially uses the GPS backed up by us verifying what it is doing through manually tuning in the required navaid such as an NDB, LOC, DME, VOR etc. As part of the approach briefing we verify the integrity of the database including the platform altitude that we start down on and any step altitudes and the distances between each fix. We are prohibited from amending the database at all as a further safety measure.
The entire procedure can be flown fully managed by the auto pilot only leaving us to disconnect once visual and land. As you rightly point out many non precision approaches are offset so we get to do a little more work to ensure the aircraft is correctly lined up with the runway and that we are stablised and configured correctly in accordance with our company SOP's.
Its a superb system that Airbus have come up with and in my opinion enhances safety to a high degree but like everything its a case of garbage in garbage out so we are very careful to make sure we know what the aircraft is doing and why.