The optimal design of refrigerators and heat pumps has been developed. The machine has two sources of irreversibilities: the thermal resistance between the machine and the hot and cold ends and the internal irreversibilities during the compression and expansion processes. The results show that there is an optimum balance between the sizes of the heat exchangers at the hot and cold ends of the machine which maximize the rate of heat removed by a refrigerator or the rate of heat delivered by a heat pump. No optimal temperature ratio across the inner compartment of the machine can be observed.