The mechanical modelling of the behaviour of materials subject to large strain is a concern in a number of engineering applications. During deformation, the material may remain in the elastic range, as, for instance, when a rubber band is stretched, but usually inelasticity is involved, as, for instance, when a metal staple is bent. The achievement of severe deformations involves the possibility of the nucleation and development of non-trivial deformation modes—including localized deformations, shear bands and fractures—, emerging from nearly uniform fields. The description of the conditions in which these modes may appear, which can be analysed through bifurcation and stability theory, represents the key for the understanding of failure of materials and for the design of structural elements working under extreme conditions. Bifurcation and instability modes occur in a variety of geometrical forms (as can be shown with the example of a cylinder subject to axial compression) and may explain the so-called ‘size effect’, ‘softening’ and ‘snap-back’ even when fracture, damage and inelasticity are excluded. Shear banding can occur as an isolated event, leading to global failure, or as a repetitive mechanism of strain ‘accumulation’ (as can be shown through the examples of chains with softening elements). Features determining bifurcation loadings and modes strongly depend on the constitutive features of the materials involved (as can be shown with the example of the Shanley model for inelastic column buckling).