Thin ribbons 30–50 μm thick of the Ti₅₀Ni_{50 –x}Cu_x (x = 25, 26, 28, 30, 32, 34; 36, 38 at %) alloys are produced by the melt-spinning technique at a cooling rate of approximately 10⁶ K/s. It is revealed that at the content of copper 25 and 26 at % the alloys are in the amorphous-crystalline state, while at the content of copper above 26 at % the alloys are in the amorphous state. It is shown that isothermal crystallization of alloys with the copper content from 25 to 32 at % leads to formation of single-phase B2 structure, which upon cooling transforms into the martensite phase B19. A noticeable decrease in characteristic temperatures of martensitic transformation at the increase in content of copper is revealed. In the alloys with the content of copper more than 32 аt % after crystallization, the two-phase structure of B2 + B11 is formed; at the same time, the B11 phase (TiCu) substantially prevents B19 ↔ B2 transformation in the alloy with 34 at % copper and blocks all structural transformations in the alloys with 36 and 38 at % copper.