Copper/diamond composites have drawn lots of attention in the last few decades, due to its potential high thermal conductivity and promising applications in high-power electronic devices. However, the bottlenecks for their practical application are high manufacturing/machining cost and uncontrollable thermal performance affected by the interface characteristics, and the interface thermal conductance mechanisms are still unclear. In this paper, we reviewed the recent research works carried out on this topic, and this primarily includes (1) evaluating the commonly acknowledged principles for acquiring high thermal conductivity of copper/diamond composites that are produced by different processing methods; (2) addressing the factors that influence the thermal conductivity of copper/diamond composites; and (3) elaborating the interface thermal conductance problem to increase the understanding of thermal transferring mechanisms in the boundary area and provide necessary guidance for future designing the composite interface structure. The links between the composite’s interface thermal conductance and thermal conductivity, which are built quantitatively via the developed models, were also reviewed in the last part.