This thesis provides a study of different aspects of the mechanical and vibrational properties of disordered and amorphous solids. Resorting to the theoretical framework of non-affine lattice dynamics the attention is focused on the analysis of disordered networks and lattices which serve as tractable model systems for real materials. Firstly, we discuss the static elastic response and the vibrational spectra of defective fcc crystals. The connection to different types of microstructural disorder in the form of bond-depletion and vacancies is described within the context of the inversion symmetry breaking of the local particle configurations. We identify the fluctuations of the local inversion symmetry breaking, which is directly linked to the non-affinity of the disordered solid, as the source of different scalings behaviours of the position of the boson peak. Furthermore, we describe the elastic heterogeneities occurring in a bond-depleted two- dimensional lattice with long-range interactions. The dependence of the concomitant correlations of the local elastic moduli are studied in detail in terms of the interaction range and the degree of disorder. An analytical scaling relation is derived for the radial part of the elastic correlations in the affine limit. Subsequently, we provide an argument for the change of the angular symmetry of the elastic correlation function which was observed in simulations and experiments on glasses and colloids, respectively. Moving to the dynamical behaviour of disordered solids, a framework is developed based on the kernel polynomial method for the approximate computation of the non- affine correlator of displacement fields which is the key requirement to describe the linear viscoelastic response of the system within the quasi-static non-affine formalism. This approach is then extended to the case of multicomponent polymer melts and validated against molecular dynamics simulations at low non-zero temperatures. We also consider the dynamical behaviour of metallic glasses in terms of its shear elasticity and viscosity. A theoretical scheme is suggested which links the repulsive strength of the interatomic potential to the viscoelasticity and fragility in metallic glasses in the quasi-affine limit.