In this paper, nuclear data for cross sections of the 64Zn(n,2n)63Zn, 64Zn(n,3n)62Zn, 64Zn(n,p)64Cu, 66Zn(n,2n)65Zn, 66Zn(n,p)66Cu, 67Zn(n,p)67Cu, 68Zn(n,p)68Cu, and 68Zn(n,α)65Ni reactions were studied for neutron energies up to 40 MeV. In the nuclear model calculations, TALYS 1.6, ALICE/ASH, and EMPIRE 3.2 codes were used. Furthermore, the nuclear data for the (n,2n) and (n,p) reaction channels were also calculated using various cross-section systematics at energies around 14–15 MeV. The code calculations were analyzed and obtained using the different level densities in the exciton model and the geometry-dependent hybrid model. The results obtained from the excitation function calculations are discussed and compared with literature experimental data, ENDF/B-VII.1, and the TENDL-2015 evaluated data.