This work deals with a comprehensive theoretical and numerical framework that allows the modeling of finite strain magnetorheological elastomers (MREs) comprising mechanically soft nonlinear elastic-viscoelastic polymer phases and magnetically hard (i.e. dissipative) or soft (i.e. purely energetic) magnetic phases. The framework is presented in a general manner and is implemented using the finite element method. Two software implementations are developed, one using FEniCS and the other in Abaqus. A detailed analysis of the numerical schemes used to model the surrounding air is made and their pros and cons are discussed. The proposed framework is used to simulate two model geometries that are directly relevant to recent applications of MREs. The first two-dimensional example simulates a mechanically soft beam consisting of a single wavy-chain of hard or soft magnetic particles. The beam is subjected to transverse magnetic actuation loads that induce important vertical deflections. Despite the overall small local strains in the beam, a significant viscoelastic effect is observed when high-rate magnetic fields are applied. A torque model for the particles is also used to analyze the beam geometry and is found to be in relatively good agreement with the rest of the approaches for small actuation fields. The second example discusses the rotation of a three-dimensional ellipsoid embedded in a cubic elastomer domain, while the ensemble lies inside a larger cubic air domain. Non-monotonic uniaxial and rotating magnetic fields are applied leading to complex, non-monotonic rotations of the ellipsoidal particle. The hard and soft magnetic cases exhibit significant differences, whereas viscoelasticity is found to induce strong coupling with the magnetization rotation but not with the dissipative magnetization amplitude. Extensive supplementary material provides all details of our implementations as well as animated visualization of results.