Recent research upon the incorporation of graphene into copper matrix composites is reviewed in detail. An extensive account is given of the large number of processing methods that can be employed to prepare copper/graphene composites along with a description of the microstructures that may be produced. Processing routes that have been employed are described including powder methods, electrochemical processing, chemical vapour deposition, layer-by-layer processing, liquid metal infiltration among a number of others. The mechanical properties of the composites are described in detail along with an account of the structural factors that control mechanical behaviour. The mechanics and mechanisms of deformation are discussed, and the effect of factors such as the graphene content and the type of graphene used, along with processing conditions for the fabrication of the composites, is described. The functional properties of copper/graphene composites are also reviewed including their electrical and thermal properties, and tribological and corrosion behaviour. In each case, the effect of the graphene type and content, and processing conditions are also described. Finally, possible future applications of copper/graphene composites are discussed.