Abstract: A sheet of glass or similar material is moved to and in the cutting area of the apparatus by sheet gripping assemblies that engage and grip an edge portion of the sheet, and that undergo precise controlled movement. A fluid-jet discharging nozzle moves above the sheet along a path of travel normal to the sheet movement. Belts supportively underlying the sheet are driven sufficiently in unison therewith to minimize possible scratching or other marring of the glass. To the same end, the lower portion of the fluid-jet nozzle is of tapered shape. Sheet-engaging clamps of the gripping assemblies are moveable in a manner adapted to prevent damage to the glass sheets engaged thereby. Positioning structure may be provided for receiving and positioning the glass sheets preparatory to the cutting thereof.