Shape memory alloys are typical temperature-sensitive metallic functional materials due to superelasticity and shape recovery characteristics. The conventional shape memory effect involves the formation and deformation of thermally induced martensite and its reverse transformation. The shape recovery process usually takes place over a temperature range, showing relatively low temperature-sensitivity. Here we report novel Cu-Al-Fe-Mn shape memory alloys. Their stress-strain and shape recovery behaviors are clearly different from the conventional shape memory alloys. In this study, although the Cu-12.2Al-4.3Fe-6.6Mn and Cu-12.9Al-3.8Fe-5.6Mn alloys possess predominantly L2(1) parent before deformation, the 2H martensite stress-induced from L2(1) parent could be retained after unloading. Furthermore, their shape recovery response is extremely temperature-sensitive, in which a giant residual strain of about 9% recovers instantly and completely during heating. At the same time, the phenomenon of the jumping of the sample occurs. It is originated from the instantaneous completion of the reverse transformation of the stabilized 2H martensite. This novel Cu-Al-Fe-Mn shape memory alloys have great potentials as new temperature-sensitive functional materials.