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Hard-magnetic soft composites consist of magneto-active polymers (MAPs) where the fillers are composed of hard-magnetic (magnetically polarised) particles. These novel multifunctional materials are experiencing a great advance from the last few years. This rise has been motivated by the possibility of controlling ferromagnetic patterns during the manufacturing process. Thus, structures with programmable functionalities can be conceptualised and implemented, opening new routes into the design of smart components with great opportunities in the biomedical engineering and soft robotics fields. In this work, we provide an overview of the state of the art of such MAPs, providing the key fundamentals and reference works. To this end, we present the current synthesis and experimental characterisation methods, the different computational modelling approaches across scales, and a detailed presentation of their current potential applications. Finally, we provide an overall discussion on future perspectives.