Ti-Ni-Cu alloys are very attractive shape memory alloys for applications as actuators because of a large transformation elongation and a small transformation hysteresis. Rapidly solidified Ti-Ni alloy ribbons have been known to have the shape memory effect and superelasticity superior to the alloy ingots fabricated by conventional casting. In this study, solidification structures and shape memory characteristics of $Ti-Ni_{30}-Cu_{20}$ alloy ribbons prepared by melt spinning were investigated by means of DSC and XRD. Operating parameters to fabricate the amorphous ribbons were the wheel velocity of 55 m/s and the melt spinning temperature of $1500^{circ}C$. The crystallization temperature was measured to be $440^{circ}C$. The crystallized ribbons exhibited very fine microstructure after annealing at $440^{circ}C$ for 10 minutes and $460^{circ}C$ for 5 minutes and was deformed up to about 6.8% and 6.23% in ductile manner, respectively. Stress-strain curve of the ribbon exhibited a flat stress-plateau at 64 MPa and this is associated with the stress-induced a B2-B19 martensitic transformation. During cycle deformation with the applied stress of 220 MPa, transformation hysteresis and elongation associated with the B2-B19 transformation were observed to be $4.3^{circ}C$ and 3.6%.