In this thesis calculations of the vibrational spectra appropriate to structural models of the elemental amorphous semiconductors a-Ge, a-As and a-Se are presented. Simple dynamical models, involving restoring forces for bond length and angle variations only, provide information on the structure-dependence of the vibrational spectra and hence on the typical structures of the real materials. Calculations are presented for four-fold coordinated continuous random network (CRN) models of a-Ge, for a three-fold coordinated CRN model of a-As and for isolated- and interacting-chain models of a-Se. In order to obtain a more realistic description of the structure and vibrational and electronic behaviour of a-Ge, calculations of the vibrational and electronic spectra of a series of seven CRS models are presented. The results show that the form of the vibrational spectrum is determined by the angular distortions whereas the form of the electronic spectrum is determined by the topology of the corresponding network. The vibrational spectra calculated using the more realistic dynamical model for a-Ge are also significantly different to those obtained previously using simple bond stretching and bending forces only. The results are discussed in relation to the structure of a-Ge and amorphous III-V compounds such as a-GaAs, the existence of an 'excess specific heat' at low temperatures in amorphous semiconductors, the applicability of such calculations to other systems and the modifications of the method for the calculations of infra-red and Raman spectra appropriate to the structural models. Simple force constant models, of the type applied to the elemental amorphous semiconductors, are also used to identify the forces chiefly responsible for the observed optical phonon anisotropies in the tin dichalcogenide layer compounds SnS2 and SnSe2.