Magnetic, magnetocaloric and critical properties of Ni50-xCuxMn37Sn13 (x = 0, 1, 2, 4 and 8) rapidly quenched ribbons have been studied. The substitution of Cu for Ni clearly effects on magnetic transitions, magnetocaloric effects and magnetic orders of these alloy ribbons. With increasing the Cu-concentration, the martensitic-austenitic phase transition shifts to lower temperature, from 265 K (for x = 0) to 180 K (for x = 4), and disappears with x = 8, while the Curie temperature, T-C, is almost unchanged. Both the conventional and inverse magnetocaloric effects are observed. The obtained values for the maximum inverse magnetic entropy changes, vertical bar Delta S-m vertical bar(max), of the ribbons are relatively large, 5.6 J kg(-1) K-1 (for x = 0) and 5.4 J kg(-1) K-1 (for x = 4) with the external magnetic field change Delta H = 12 kOe. The critical parameters (T-C, beta, gamma and delta) of the ribbons are determined from the static magnetic data at the second order ferromagnetic-paramagnetic transition by using both the Arrott-Noakes and Kouvel-Fisher methods. The results reveal that the long-range ferromagnetic order in the alloy is tendentiously dominated by increasing the Cu-concentration. (C) 2014 Elsevier B.V. All rights reserved.