Measurements of the spectra of neutrons moderated in LiH were made in the energy range of about 0.01 to 600 eV, and the results were compared with calculated spectra obtained from a Monte Carlo calculation, a direct numerical integration of the Boltzmann equation (NIOBE code), a moments numerical calculation, and three infinite-medium thermalization calculations, each utilizing a different scattering kernel. The measurements were carried out by irradiating slabs of LiH with neutrons having a near-fission spectrum. The spectra of the leakage flux, of the forward-directed flux, and of the scalar flux within the slab were obtained at neutron penetrations of 2.5 to 10 cm. Below 30 eV, the leakage flux and scalar flux attained an asymptotic spectral shape at a penetration of 2.5 cm, and the forward-directed flux at about 5 cm. The shapes of the calculated spectra agree with the shapes of the measured spectra for all energy regions in which each calculation is valid. A large discrepancy between the NIOBE code predictions and the measurements below 0.08 eV is caused by upscattering and molecular binding effects, which are neglected by NIOBE. These effects were included in a neutron thermalization calculation for an infinite medium with a constant source density; however, good agreement with measurement was obtained only for the case in which the measurement had been made in a nearly gradient-free region. In a region of strong flux gradients, the spectrum of the forward-directed flux is shown to be related to that of the scalar flux with good accuracy by the Purohit expression, according to a NIOBE code calculation which yielded both spectra.