The chapter discusses the localization of plastic deformation into shear bands, which is often a precursor to rupture. It reviews experimental observations, presents a general theoretical framework, and performs specific calculations for various material models. The interplay between inelastic constitutive descriptions, such as deviations from normality and vertex-like yielding, and the onset of localization is emphasized. The introduction highlights the common occurrence of localization in ductile solids, including metals, polymers, and geological materials, and notes the lack of a comprehensive understanding of the phenomenon. Examples of localization are provided, such as Lüders band formation in metals and localized deformation in ductile metal polycrystals. The mechanisms of localization are explored, including the role of rate-independent and thermally decoupled constitutive models. The general theory is developed, considering deformations and stress rates, and the conditions for static localization are discussed. The chapter also examines the acceleration waves and dynamic growth of disturbances, the response with initial imperfections, and the limiting nature of the localization instability. Finally, specific results for rigid-plastic materials, elastic-plastic materials, and plastically dilatant materials with pressure-sensitive yielding are presented.The chapter discusses the localization of plastic deformation into shear bands, which is often a precursor to rupture. It reviews experimental observations, presents a general theoretical framework, and performs specific calculations for various material models. The interplay between inelastic constitutive descriptions, such as deviations from normality and vertex-like yielding, and the onset of localization is emphasized. The introduction highlights the common occurrence of localization in ductile solids, including metals, polymers, and geological materials, and notes the lack of a comprehensive understanding of the phenomenon. Examples of localization are provided, such as Lüders band formation in metals and localized deformation in ductile metal polycrystals. The mechanisms of localization are explored, including the role of rate-independent and thermally decoupled constitutive models. The general theory is developed, considering deformations and stress rates, and the conditions for static localization are discussed. The chapter also examines the acceleration waves and dynamic growth of disturbances, the response with initial imperfections, and the limiting nature of the localization instability. Finally, specific results for rigid-plastic materials, elastic-plastic materials, and plastically dilatant materials with pressure-sensitive yielding are presented.