  This review presents a unified view on the problem of Anderson localization
in one-dimensional weakly disordered systems with short-range and long-range
statistical correlations in random potentials. The following models are
analyzed: the models with continuous potentials, the tight-binding models of
the Anderson type, and various Kronig-Penney models with different types of
perturbations. Main attention is payed to the methods of obtaining the
localization length in dependence on the controlling parameters of the models.
Specific interest is in an emergence of effective mobility edges due to certain
long-range correlations in a disorder. The predictions of the theoretical and
numerical analysis are compared to recent experiments on microwave transmission
through randomly filled waveguides.
