Abstract: The one-way and two-way shape memory effects as well as the superelasticity and the strain variation on cooling and heating under stress were studied in the cast Ti40.7Hf9.5Ni49.8 – xCux (x = 1, 5, 10 at %) alloys. It was found that after preliminary deformation by tension in martensite state, the maximum value of shape memory effect was close to 4% for the alloys with x = 1 and 5 at % and 5.3% for the alloy with x = 10 at %. The maximum value of the two-way shape memory effect was equal to 2% and hardly depended on the Cu concentration in the alloys. This value was several times higher than the values previously observed in the quaternary Ti–Hf–Ni–Cu alloys. The maximum value of the recoverable strain observed on cooling and heating under constant stress did not depend on the copper concentration in the alloy and it was close to 6%. In the alloys with x = 1 and 5 at %, the recovery coefficient after cooling under stress was higher than after active deformation in the martensite state. In the alloy with x = 10 at %, the recovery coefficient was not affected by the way of the preliminary deformation. It was found that an increase in the copper concentration increased the volume fraction of martensite transforming into austenite on unloading, however the strain recovery was not complete on unloading. This was due to the manifestation of the martensite stabilization effect and that the critical stress for martensite nucleation was close to the yield limit for dislocation slip in the studied alloys.