A detailed understanding of fundamental material properties can be obtained through the study of atomic vibrations, performed experimentally with neutron scattering techniques and coupled with the two powerful new computational methodologies I have developed. The first approach involves phonon-based simulations of the pair distribution function -- a histogram of localised atomic positions generated experimentally from total scattering data. This is used to reveal ordering behaviour, to validate interatomic models and localised structure, and to give insights into how far dynamic behaviour can be studied using total scattering techniques. Most importantly, the long-standing controversy over dynamic disorder in $\beta$-cristobalite is resolved using this technique.

Inelastic neutron spectroscopy (INS) allows \emph{direct} study of vibrational modes through their interaction with the neutron beam, and is the experimental basis for the second strand of the new methodology. I have developed new simulation and refinement tools based on the next generation of spectrometers currently being commissioned at the ISIS pulsed neutron source. This allows a detailed powder spectroscopy study of cristobalite and vitreous silica demonstrating that the Bose peak and so-called `fast sound' features can be derived from standard lattice dynamics in both the crystal and the amorphous counterpart, and allowing discussion of their origins. Given the controversy in the literature, this is a key result.

The new methodology also encompasses refinement of interatomic models against powder INS data, with aluminium providing a successful test-case. A more complex example is seen in calcite, with experimental data collected during the commissioning of the new MERLIN spectrometer. Simulated one-phonon coherent INS spectra for the single crystal and powder (the latter including approximations to multi-phonon and multiple scatter) are fully convolved with experimental resolution functions. These are used in the analysis of the experimental data, yielding previously unpublished dispersion curves and soft mode information, as well as allowing the effectiveness of powder refinement of more complex materials to be assessed.

Finally, I present further applications with technologically important materials -- relaxor ferroelectrics and high temperature pnictide superconductors. The conclusions draw together the different strands of the work, discussing the importance of these new advances together with future developments and scientific applications.