An overview is presented on the structural and phase transformations and physical and mechanical properties of those multicomponent copper-based shape memory alloys which demonstrate attractive commercial potential due to their low cost, good shape memory characteristics, ease of fabrication, and excellent heat and electrical conductivity. However, their applications are very limited due to brittleness, reduced thermal stability, and mechanical strength—properties which are closely related to the microstructural features of these alloys. The efforts of the authors of this article were aimed at obtaining a favorable microstructure of alloys using new alternative methods of thermal and thermomechanical treatments. For the first time, the cyclic martensitic transformations during repeated quenching, methods of uniaxial megaplastic compression, or torsion under high pressure were successfully applied for radical size refinement of the grain structure of polycrystalline Cu-Al-Ni-based alloys with shape memory. The design of the ultra- and fine-grained structure by different methods determined (i) an unusual combination of strength and plasticity of these initially brittle alloys, both under controlled heat or hot compression or stretching, and during subsequent tensile tests at room temperature, and, as a consequence, (ii) highly reversible shape memory effects.