Abstract: A machine for grinding the rotor blades (4) of an aeronautical engine is equipped with a CNC control and its header (1) is a rotatable header provided with at least two grinders (2) of different abrasive materials, wherein each of them is coupled to the header (1) for being selected and then positioned in contact with the different stage blades (4) with the cooperation of the CNC. The machine for grinding also includes a measuring unit (10) fixed to the machine bed for controlling the radius of the blade stage, slides (11, 13) to displace the header (1) in two directions (X, W), and equipment to rotate the header (1) an angle (C) about its central transverse axis (1a). The CNC has registered the values of the grinder diameters (D1, D2) and the fixed position data (R1, A, R2, B) of the respective grinder profile (14, 15) relative to the central axis (1a).