Abstract: A machine for flatbed cutting of pieces from a strip of material (3) wound in a coil and moving continuously. The cutting zone (ZD) comprises a cutting head (17) supported by a mobile support (20) translatably mounted on a stationary longitudinal beam (18) disposed parallel to the direction of movement (A) of the strip of material. It is coupled to a first actuator (Mx) for moving the cutting head (17) linearly along a longitudinal axis (X) parallel to the beam and comprises a variable-geometry mechanism (21-24) coupled to a second actuator (My) for moving the cutting head (17) linearly along a transverse axis (Y), perpendicular to the longitudinal axis (X). A central control unit (9) controls the actuators to move the cutting head (17) in the cutting zone (ZD) along a path determined by the contour of the pieces (2) to be cut and the speed of the material (3).