The paper revisits the central role and properties of the notion of tangent modulus tensor in finite-strain elastoplasticity. Here, in the current configuration of continuum mechanics, a compact expression for this tensor is derived for materials experiencing finite strain due to plastic flow, starting from yield and flow relations referred to the current configuration. The authors address issues posed by the celebrated decomposition into elastic and plastic strains and by the objective stress flux. Both ``associated'' and ``non-associated'' models can be accommodated, as the so-called ``plastic incompressibility''. A constitutive inequality with uniqueness implications is formulated which extends the condition for ``stability in the small'' to the finite-strain setting. The authors propose modifications of the tangent modulus tensor that accommodate kinematic hardening. This is implemented in an application dealing with the finite torsion of a shaft comprised of steel described by means of a von Mises yield criterion with isotropic hardening.