This paper presents an overview of research efforts focused on both fabrication and mechanical properties of copper-matrix nanocomposites containing graphene nanoinclusions. A particular attention is devoted to experimental data giving evidence for enhancement of strength, microhardness and Young modulus (due to the effects of graphene nanoinclusions) of such nanocomposites, as compared to unreinforced copper. Key factors are discussed which influence the role of graphene nanosheets and nanoplatelets in remarkable enhancement of the mechanical characteristics of copper-graphene nanocomposites.