The invention discloses a magneto-rheological torsion damper of a clutch and a clutch with the same. The magneto-rheological torsion damper comprises a sleeve and a torque input part, wherein a containing cavity is formed in the sleeve, and the torque input part is fixedly connected with the sleeve and inputs torque. The magneto-rheological torsion damper further comprises an elastic vibration damping element and a magneto-rheological damper part, wherein the elastic vibration damping element can buffer vibration through elastic compression when the torque is input, the magneto-rheological damper part rotates inside the annular containing cavity to generate magneto-rheological damping force under the force transmission of the elastic vibration damping element, and the annular containing cavity is filled with magneto-rheological fluid media. The magneto-rheological torsion damper has the advantages of being compact in structure, high in adaptability, small in energy dissipation and the like, can achieve the functions of a traditional torsion damper of the clutch, and meanwhile can solve the problem that damping parameters of an existing driven disc type torsion damper are uncontrollable.