Matera's old town was abandoned in the 1950s when the Italian government used force to relocate the city's inhabitance. But due to tourism some select places the scavi have been reinhabited by hotels, restaurants and souvenir shops. The modern city with car friendly streets are where people live and work now surrounds the old town. Its just the steep narrow paths on the hill slope that has been abandoned. The caves can be seen from the old city.
Click (here) for a map of Matera.

The old caves with the modern city above