Magneto-active polymers (MAPs) consist of a polymeric matrix filled with magnetisable particles. MAPs may change their mechanical properties (i.e., stiffness) and/or mechanical deformation upon the application of an external magnetic stimulus. Mechanical responses of MAPs can be understood as the combined contributions of both polymeric matrix and magnetic particles. Moreover, the magnetic response is defined by the interaction between magnetisable particles and the external field. Common approaches to model MAPs are based on phenomenological continuum models, which are able to predict their magneto-mechanical behaviour but sometimes failed to illustrate specific features of the underlying physics. To better understand the magneto-mechanical responses of MAPs and guide their design and manufacturing processes, this contribution presents a novel continuum constitutive model originated from a microstructural basis. The model is formulated within a finite deformation framework and accounts for viscous (rate) dependences and magneto-mechanical coupling. After the formulations, the model is calibrated with a set of experimental data. The model is validated with a wide range of experimental data that show its predictability. Such a microstructurally-motivated finite strain model will help in designing MAPs with complex three-dimensional microstructures. (C) 2020 The Authors. Published by Elsevier Ltd.