This paper focuses on the numerical analysis of quasi-static viscoplastic and elastoplastic evolution problems with isotropic or kinematic hardening. The authors present a refined a priori and a posteriori error analysis for the spatial discretization, emphasizing optimal convergence and adaptive mesh-refining. Within each time-step of an implicit time-discretization, the finite element method leads to a minimization problem for non-smooth convex functions on discrete subspaces. For piecewise constant or affine ansatz functions, the stress and displacement approximations are shown to converge linearly. An a posteriori error estimate justifies an automatic adaptive mesh-refining algorithm, which is supported by numerical examples demonstrating its superiority over non-adapted meshes. The paper also discusses the dual stress formulation and its comparison with the primal formulation, highlighting the advantages of the primal formulation in terms of higher-order schemes. The analysis includes a detailed error analysis and numerical solution algorithms, including a quasi-Newton-Raphson scheme for efficient computation.This paper focuses on the numerical analysis of quasi-static viscoplastic and elastoplastic evolution problems with isotropic or kinematic hardening. The authors present a refined a priori and a posteriori error analysis for the spatial discretization, emphasizing optimal convergence and adaptive mesh-refining. Within each time-step of an implicit time-discretization, the finite element method leads to a minimization problem for non-smooth convex functions on discrete subspaces. For piecewise constant or affine ansatz functions, the stress and displacement approximations are shown to converge linearly. An a posteriori error estimate justifies an automatic adaptive mesh-refining algorithm, which is supported by numerical examples demonstrating its superiority over non-adapted meshes. The paper also discusses the dual stress formulation and its comparison with the primal formulation, highlighting the advantages of the primal formulation in terms of higher-order schemes. The analysis includes a detailed error analysis and numerical solution algorithms, including a quasi-Newton-Raphson scheme for efficient computation.