A new formulation of the knock-out mode in (d, p) or (d, n) reactions is attempted with the cut-off Born approximation and the j- j coupling scheme for the initial and final nuclei. During the collision between an incident deuteron and the target proton (or neutron), the proton-proton (or neutron-neutron) interaction as well as the neutron-proton interaction are actually taken into consideration. The theoretical analyses are carried out on the reactions of N 14(d, p)N 15(g.s.), C 12(d, p)C 13(g.s.) and Be 9(d, p)Be 10(g.s.) at several incident energies, with the assumption that the reactions take place only through the stripping and knock-out modes. Some systematic deviations of the experimental cross-sections from the simple Butler theory, observed in the range of E d = 3 to 15 MeV, can be explained satisfactorily through allowing for knock-out effects. Under the same assumption, the O 16(d, n)F 17(g.s.) reaction together with the O 16(d, p)O 17(g.s.) reaction are studied for several deuteron energies with satisfactory results. The Mg 24(d, p)Mg 25(g.s.) reaction is also similarly analysed. The anomaly of the angular distribution observed at E d = 14.8 MeV in this reaction is discussed in some detail. The angular distributions of the B 10(d, p)B 11∗(2.14 MeV) reaction in the range E d = 8.1 to 10.0 MeV are calculated as a pure knock-out mode and the results are compared with those of experiment and previous theories. The result of this analysis is found to favour the present theory.