Abstract: A known type of heat pump or refrigeration apparatus comprises a closed circuit containing a refrigerant, the closed circuit comprising an acceptor for heat exchange between the refrigerant and a first body of a fluid or other substance, a compressor for compressing the refrigerant from the acceptor, a rejector for heat exchange between the compressed refrigerant and a second body of a fluid or other substance, and an expansion device to expand the refrigerant from the rejector before it is directed back to the acceptor. In the known apparatus, the refrigerant is at subcritical pressure at all places in the closed circuit. In contrast, in the circuit of apparatus embodying the invention, whereas the refrigerant in the acceptor remains at a subcritical pressure, the refrigerant in the rejector is at supercritical pressure. This enables the entropy gain in the rejector to be substantially reduced and the thermodynamic efficiency (and also the coefficient of performance) to be increased.