Machine for grinding and/ or polishing slabs of stone material, such as natural or agglomerated stone, ceramics or glass, comprising: a support bench for the slabs to be machined; there being provided above the support bench : at least one machining station comprising at least one pair of bridge support structures situated opposite each other and arranged transversely on either side of the support bench , first means for performing a relative constant movement in the longitudinal direction of machining station and slab on the support bench , and at least one beam , the two ends of which are supported by said support structures ; at least one rotating spindle with a sliding vertical axis mounted on said at least one beam , said beam being movable transversely on said support structures , the bottom end of the spindle being provided with at least one tool-holder support rotating about the axis of rotation of said spindle and carrying at least one abrasive tool for forming the grinding and/ or polishing heads; second means for relative movement of said at least one spindle with respect to the support bench in the longitudinal direction; and a programmable control unit for adjusting the alternating movement in the transverse direction of the beam and the alternating longitudinal movement due to the second means by operating the respective movement means for relative movement of the at least one spindle; characterized in that the programmable control unit is designed for adjusting the movement of the first relative movement means and in that the first relative movement means consist of a conveyor belt which causes feeding of the slab with a constant movement so as the slab being machined moves with relative constant movement underneath the machining station over its entire length. at least one machining station comprising at least one pair of bridge support structures situated opposite each other and arranged transversely on either side of the support bench , first means for performing a relative constant movement in the longitudinal direction of machining station and slab on the support bench , and at least one beam , the two ends of which are supported by said support structures ; at least one rotating spindle with a sliding vertical axis mounted on said at least one beam , said beam being movable transversely on said support structures , the bottom end of the spindle being provided with at least one tool-holder support rotating about the axis of rotation of said spindle and carrying at least one abrasive tool for forming the grinding and/ or polishing heads; second means for relative movement of said at least one spindle with respect to the support bench in the longitudinal direction; and a programmable control unit for adjusting the alternating movement in the transverse direction of the beam and the alternating longitudinal movement due to the second means by operating the respective movement means for relative movement of the at least one spindle; characterized in that the programmable control unit is designed for adjusting the movement of the first relative movement means and in that the first relative movement means consist of a conveyor belt which causes feeding of the slab with a constant movement so as the slab being machined moves with relative constant movement underneath the machining station over its entire length. <EOS>