  In glasses, atomic disorder combined with atomic connectivity makes
understanding of the nature of the vibrations much more complex than in
crystals or molecules. With a simple model, however, it is possible to show how
disorder generates quasi-local modes on optic branches as well as on acoustic
branches at low-frequency. The latter modes, possibly hybridizing with
low-lying optic modes in real glasses, lead to the excess, low-frequency
excitations known as {\it boson-peak modes}, which are lacking in crystals. The
spatially quasi-localized vibrations also explain anomalies in thermal
conductivity and the end of the acoustic branches, two other specific features
of glasses. Together with the quasi-localization of the modes at the nanometric
scale, structural disorder lifts the crystalline or molecular spectroscopic
selection rules and makes interpretation of experiments difficult.
Nevertheless, vibrations in simple glasses such as vitreous silica or vitreous
boron oxide are nowadays rather well described. But a comprehensive
understanding of the boson peak modes remains a highly debated issue as
illustrated by three archetypal glass systems, vitreous SiO$_2$ and B$_2$O$_3$
and amorphous silicon.

Disordered Systems and Neural Networks