Neutron emission spectrometry (NES) measures the energies of neutrons produced in fusion reactions. Here we present velocity-space weight functions for NES and neutron yield measurements. Weight functions show the sensitivity as well as the accessible regions in velocity space for a given range of the neutron energy spectrum. Combined with a calculated fast-ion distribution function, they determine the part of the distribution function producing detectable neutrons in a given neutron energy range. Furthermore, we construct a forward model based on weight functions capable of rapidly calculating neutron energy spectra. This forward model can be inverted and could thereby be used to directly measure the fast-ion phase-space distribution functions, possibly in combination with other fast-ion diagnostics. The presented methods and results can be applied to neutron energy spectra measured by any kind of neutron spectrometer and to any neutron yield measurement.