Neutron vibrational spectroscopy (inelastic neutron scattering (INS)) is a form of vibrational spectroscopy that is complementary to infrared and Raman spectroscopies. It has a very wide spectral range of zero to several thousand wavenumbers; there are no selection rules, so all modes are allowed and comparison with ab initio calculations is straightforward. The disadvantages are that it is relatively insensitive and restricted to the solid state. It is particularly sensitive to modes involving hydrogen, and in hydrogenous materials, these dominate the spectra. In this article, the history, theory, and practice of the technique are briefly reviewed. The extensive range of applications of this form of spectroscopy is illustrated with examples in the areas of hydrogen storage, catalysis, soft condensed matter, ice, carbons, and magnetism.