A machine for smoothing or polishing slabs of stone material, such as natural and agglomerated stone, ceramic and glass, comprises a longitudinal bench (12) over which the slabs to be machined move, at least one pair of opposite bridge support structures (20, 22) arranged astride the bench, and at least one beam (24) transversally movable and supported by said bridge structures. At least one vertical-axis and vertically movable mandrel (40) is mounted eccentrically on a mandrel supporting structure (30) which rotates about its vertical axis (Z1) and is supported on said beam (24). The mandrel has, mounted on its bottom end, a device carrying the smoothing or polishing tools and rotating about the axis of rotation of said mandrel. In this way, the tool carrying device performs a movement composed of the rotation about the axis of rotation of the mandrel, a revolving movement about the axis of rotation of the mandrel carrying structure and a translation movement due to the movement of the beam.