Abstract: Apparatus for cutting or scoring sheet material, e.g. glass, with rapid responses to irregularities in the surface of the sheet material and to commands to commence or interrupt a cutting or scoring operation, employs a fluid pressure actuated diaphragm device to impart movement to a cutting or scoring tool, e.g. a cutting wheel, towards and away from the surface to be cut or scored, and means for controlling selectively the pressure acting on each side of the diaphragm. The tool may be supported and guided in a cutter head which incorporates the diaphragm device and which carries supporting means, such as a pair of wheels, for supporting the cutter head on the surface of the material when the tool is retracted. Fluid pressure supply to the diaphragm device and to a separate piston device for applying the cutting load can be controlled by solenoid-operated values. Supply of a cutting fluid to the tool can be similarly controlled.