To our knowledge, the effect of Be and Y on the martensite transformation parameters in TiNi-based shape memory alloys was studied occasionally and mainly on a level of the technological additions. Recently, due to the development of non-conventio nal production technologies applied to the shape memory materials, both metals are found to be the appropriate canditates for production of the potential TiNi-based shape memory alloys in an initially amorphous structural state by means of the melt-spinning and injection casting techniques. First of all, the Be and Y attract attention due to satisfaction of the so-called golden rules known for the bulk amorphous alloy systems. We present the result of the experimental study regarding the systematic substitution of the main constituent elements Ti and Ni by Be and Y up to (2-10) at.%. In as-cast condition, all the Ti-Ni-Be and Ti-Ni-Y alloys undergo the dual B2 R B19' martensite transformation sequence, well resolved on the cooling run and often overlapping on the heating run. Both the TR and Ms transformatio n temperatures smoothly decrease with the Be or Y content, the slope to be higher for substitution of the Ti than the Ni. The ternary TiNiBe and TiNiY melt- spun ribbons produced in the standard technological condition are rather brittle and don't show a good metallic aspect. They are in the crystalline structural state and undergo the martensite transformation sequence similar to the bulk material but in the lower temperature range. The contribution of the chemical composition, microstructure and strain/stress field (caused by the difference in the atomic radii) to the martensite transformation temperatures is considered.