The exact formulation of the subloading surface model is provided in this article for the description of the elastoplastic and elasto-viscoplastic deformations not only for the monotonic but also the cyclic loading processes at general deformation rate ranging from the quasistatic to the impact loading in a unified manner by the subloading-overstress model. Here, it is noteworthy that even the elastoplastic deformation can be described more exactly by the present elasto-viscoplastic constitutive equation, noting that the elastoplastic constitutive equation is limited to the description of the quasistatic deformation behavior, but the purely quasistatic deformation does not exist actually. Therefore, the elastoplastic constitutive equation can be disused only by using the subloading-overstress model. It will be extended to describe the temperature-dependence for metals, since the elasto-viscoplastic deformation behavior is influenced by the temperature in general. Then, the validity of the extended subloading-overstress model for the prediction of the temperature-dependent elasto-viscoplastic deformation of metals will be verified by the comparisons with the test data of metals for various isothermal and/or non-isothermal deformations in the monotonic and the cyclic loading processes.