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Magnetorheological (MR) foams represent a class of smart materials with unique tunable viscoelastic properties when subjected to external magnetic fields. Combining porous structures with embedded magnetic particles, these materials address challenges such as leakage and sedimentation, typically encountered in conventional MR fluids while offering advantages like lightweight design, acoustic absorption, high energy harvesting capability, and tailored mechanical responses. Despite their potential, challenges such as non-uniform particle dispersion, limited durability under cyclic loads, and suboptimal magneto-mechanical coupling continue to hinder their broader adoption. This review systematically addresses these issues by evaluating the synthesis methods (ex situ vs. in situ), microstructural design strategies, and the role of magnetic particle alignment under varying curing conditions. Special attention is given to the influence of material composition-including matrix types, magnetic fillers, and additives-on the mechanical and magnetorheological behaviors. While the primary focus of this review is on MR foams, relevant studies on MR elastomers, which share fundamental principles, are also considered to provide a broader context. Recent advancements are also discussed, including the growing use of artificial intelligence (AI) to predict the rheological and magneto-mechanical behavior of MR materials, model complex device responses, and optimize material composition and processing conditions. AI applications in MR systems range from estimating shear stress, viscosity, and storage/loss moduli to analyzing nonlinear hysteresis, magnetostriction, and mixed-mode loading behavior. These data-driven approaches offer powerful new capabilities for material design and performance optimization, helping overcome long-standing limitations in conventional modeling techniques. Despite significant progress in MR foams, several challenges remain to be addressed, including achieving uniform particle dispersion, enhancing viscoelastic performance (storage modulus and MR effect), and improving durability under cyclic loading. Addressing these issues is essential for unlocking the full potential of MR foams in demanding applications where consistent performance, mechanical reliability, and long-term stability are crucial for safety, effectiveness, and operational longevity. By bridging experimental methods, theoretical modeling, and AI-driven design, this work identifies pathways toward enhancing the functionality and reliability of MR foams for applications in vibration damping, energy harvesting, biomedical devices, and soft robotics.