Summary: The article presents a constitutive framework of large-strain elastoplasticity in both the Lagrangian and the Eulerian geometric setting which takes into account anisotropic material response. In summary, the key ingredients of this framework are: (i) the introduction of a plastic metric which is assumed to describe locally the history-dependent inelastic material response in the sense of an internal variable formulation. (ii) The definition of a convex elastic domain in the space of the local stress-like variable conjugate to the plastic metric, denoted as the plastic force. (iii) An equivalent Lagrangian and Eulerian representation of all constitutive functions as isotropic tensor functions in terms of an extended set of arguments, denoted as anisotropy variables. (iv) The setup of normality rules for the evolution of the plastic metric and the anisotropy variables, yielding a canonical symmetric form of the elastoplastic tangent moduli. (v) A geometrically exact decomposition of the set of constitutive equations into possibly decoupled volumetric and isochoric contributions. Applications of the constitutive framework are demonstrated by means of several conceptual model problems which cover isotropic, initial anisotropic and induced anisotropic elastoplastic response.