The copper-based alloys with high strength and low electrical resistivity are of interest for many practical applications. In this study, Cu / Mg-composites with 1, 7 and 49 magnesium filaments in Cu-matrix have been obtained by hyrdoextrusion at room temperature. The structure, mechanical and electrical properties of the composite rods and thin wires in the deformed and annealed states have been investigated. The strength of the deformed Cu / Mg-composite with a minimal volume fraction of magnesium was found to be abnormally high. The increase of magnesium microhardness was revealed to be near the interface. A change in the lattice constant of the Cu-matrix in the deformed Cu / Mg-composites is discovered by the XRD-method. It has been concluded that, under severe plastic deformation, different solid solutions form on the Cu / Mg-interfaces due to mechanical alloying processes. The Cu / Mg-composite with a maximal volume fraction of Mg has the lowest strength in the deformed state, however, it becomes the strongest one after annealing at 200°C due to annealing hardening of magnesium. The temperature dependences of the electrical resistivity of Cu / Mg-composites differ significantly from each other. During heating, three eutectic transformations are realized in the composite with 7 Mg-filaments. As a result, the electrical resistivity of the deformed Cu / 7Mg-composite increases almost two times. The specific electrical resistivity of the composite with 49 Mg-filaments is a bit different from the electrical resistivity of deformed copper. The mechanisms of structure formation during the heating of these composites have been shown to be significantly different. The obtained results can be used for the development of high-strength Cu-based conductors.