The possibility of localization of a buckling pattern is investigated for a class of structures in which the initial buckling mode is periodic. A simple model indicates that the basic mechanism of localization involves a bifurcation at the maximum load point. This model also illustrates a clear analogy between localization of a buckling pattern in a structure under compressive loading and the phenomenon of necking in a bar under tensile loading. An analysis of the bifurcation that leads to localization, carried out for the more realistic structural model of a column on a softening foundation, demonstrates a delay between the maximum load point and the bifurcation point. A finite-element analysis of an elastic-plastic plate strip under axial compression shows the development of localization for a low hardening material, whereas localization does not occur in a plate made of a high hardening material for which no maximum load is reached.