The machine (10) for fitting/removing a tyre includes: a bedplate (12) for resting on the ground; a spindle (16), mounted on the bedplate and configured to tighten a rim of a wheel thereon; a demounting tool, mounted on the bedplate and able to remove a tyre from the rim when the wheel is loaded on the spindle; a bead breaking arm (30), movably connected to the bedplate and having a bead breaking tool (32) able to bead break the tyre from the rim; a lifting member (52, 54, 56), movably connected to the bedplate and able to lift the wheel to load it on the spindle after bead breaking; and an operating device (40) which is selectively connectable either to the bead breaking arm or to the lifting member and able to operate the bead breaking arm or the lifting member according to choice.