Neutrons for fusion applications stem not only from monoenergetic sources but also from “white” neutron sources. In this regard, the reaction 3H(t,n) is of particular interest. Continuous neutron spectra of the 3H(t,n) reactions were measured at 5.98-, 7.47-, 10.45-, 16.41-, and 19.14-MeV triton energy at typically seven angles between 0 and 145 deg. The spectra at the three lowest energies contain only neutrons from 3H(t,n)5He and 3H(t,2n)4He reactions and therefore can more easily be interpreted than the spectra at 16.41 and 19.14 MeV, which are too complex to allow a straightforward decomposition except for estimation of the neutron emission cross section following the reaction 3H(t,d)4H. Angle-dependent double-differential and neutron energy–integrated cross sections are given at the five energies. In most cases the peak of the two-body ground state transition could be deconvolved reliably resulting in cross sections of the reaction 3H(t,n)5He. Although the basic scale uncertainty is <5%, severe background, in particular, at the higher triton energies, increases the total uncertainty of integrated cross sections up to 9%. Naturally, the uncertainty of each energy bin of the double-differential cross sections, which depend on bin width, is considerably higher. As no previous data are reported at or near these energies, no direct comparison with other data was feasible. Evidence is provided of the formation of the short-lived neutron-rich nuclei 5He and at higher energies of 4H.