The General Atomic neutron-velocity selector has been used at the electron linear accelerator to study the inelastic scattering by liquid methane and liquid parahydrogen of monoenergetic neutrons with incident energies in the range 0.009 to 0.17 eV. The energy dependence of the total cross sections and the neutron spectra produced by specimens of these materials have also been measured. The inelastic scattering of slow neutrons (< 0.010 eV) at 90° by liquid parahydrogen appears to be smaller than expected on the basis of the measured total cross section and the angular dependence calculated by Sarma. Perhaps this is related to the fact that the total cross section is larger than for freely rotating molecules, indicating the possible existence of some hindrance to molecular motion. The slowing-down power, σnE0/E, a quantitative measure of the neutron-moderating ability, is evaluated from the measured inelastic neutron-scattering data and compared for various neutron energies for the two liquids. A consideration of the various data leads to the conclusion (1) that solid methane is better than liquid parahydrogen for production of very “cold” neutrons (E0 < 0.007 eV), and (2) that parahydrogen is superior to liquid methane for production of cold neutrons with E0 < 0.005 eV.