A method and apparatus for mounting and tightening an open clamp about an object to be fastened in which a first jaw-like member causes the band end portion intended to constitute the inner band portion to be placed about the object to be fastened, a second jaw-like member then causes the other band end portion of the clamp intended to constitute the outer band portion to be placed about the object to be fastened in such a manner that the outwardly extending hook or hooks in the inner band portion is or are adapted to engage with the corresponding aperture or apertures in the outer band portion. Two further jaw-like members each having pincer-like nose portions at the free ends are then actuated to engage with the outwardly extending leg portions of a plastically deformable ear in the clamping band to plastically deform the latter after the mechanical connection by engagement of the outwardly extending hook or hooks in the corresponding aperture or apertures is established, to tighten the clamp about the object to be fastened. A holding device thereby holds the clamp in position until the first and second jaw-like members have been actuated and assume the holding function to permit release of the clamp by the holding device. Operation of the machine is fully automatic by appropriate sequence controls of fluid-actuated piston and cylinder units.