  In this paper we investigate the spectral and the scattering theory of
Schr\"odinger operators acting on perturbed periodic discrete graphs. The
perturbations considered are of two types: either a multiplication operator by
a short-range or a long-range function, or a short-range type modification of
the measure defined on the vertices and on the edges of the graph. Mourre
theory is used for describing the nature of the spectrum of the underlying
operators. For short-range perturbations, existence and completeness of local
wave operators are also proved.
