Abstract: The present invention relates to a machine for finishing cast or machined parts comprising a closed enclosure (10) containing at least one finishing unit (14, 16) adapted to be a unit of smoothing of rough edges, of brushing, of rubbing, of polishing or of grinding and a manipulator robot (13) of the hanging type comprising a moveable arm (24) for bringing the parts in contact with the finishing tools. Aspiration elements (30, 31) are associated with each finishing unit. The finishing units are equipped with a compensation apparatus of their wear.