Extensive inelastic-neutron-scattering experiments have been performed on superfluid helium over a wide range of energy and momentum transfers. A high-resolution study has been made of the pressure dependence of the single-excitation scattering at the first maximum of the dispersion curve over the range 1-24.4 atm and at 1 atm of the wave-vector dependence of the single-excitation scattering beyond the roton minimum. The latter data, but not the former, are consistent with the idea of an instability threshold suggested by Pitaevskii and by Enz. The pressure and wave-vector dependence of the multiexcitation scattering was also studied. It is shown that the multiphonon spectrum of a simple Debye solid with the phonon dispersion and single-excitation cross section of superfluid helium qualitatively reproduces these data.