The methods of spectral analysis are of special interest in the treatment of dynamical systems, in particular, but not exclusively, when they are linear. We shall first show that by considering the spectral content of excitation and natural modes of vibration (in short the "modal spectrum") it becomes possible to identify those modes which have to be retained in the dynamical model and those which can be neglected. This aspect of modelling is of crucial importance since the modal sequence of real material systems is a priori infinite, as detailed in Volume 2. On the other hand, for many applications in mechanical engineering the pertinent information is contained in the spectral properties of the response signals and little would be gained by embarking on solving the problem in the time-domain. Finally, even if the dynamical system is far from linear, spectral analysis still provides quite valuable information concerning the existence and the nature of nonlinearities present in the system on one hand, and the periodic or the chaotic properties of the response to periodic excitation, on the other.