A normothermic perfusion device to maintain in optimal physiological conditions a liver or kidney intended to be transplanted, wherein said device comprises a permanent part and a disposable part , wherein: - the permanent part comprises: - pumps, heating elements and a control system, configured to regulate fluids in a hydraulic circuit and the temperature of said fluids, whether the container houses a liver or a kidney, and - the disposable part comprises: - a container that in turn comprises: - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container , and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container , characterized in that the central portion comprises a lower section which has a height difference with respect to an upper section and the liver or kidney is arranged between said upper section and the lower section , so that the perfusion fluid after circulating through the liver or kidney falls into the lower section , said lower section thus acting as a reservoir for perfusion fluid, and the perfusion device further comprises a connection system connected right at the inlet of the artery of the liver or kidney and fluidly communicated to the first tube and to the second tube , thus enabling the vasodilator fluid to be introduced directly into the artery. - the permanent part comprises: - pumps, heating elements and a control system, configured to regulate fluids in a hydraulic circuit and the temperature of said fluids, whether the container houses a liver or a kidney, and - pumps, heating elements and a control system, configured to regulate fluids in a hydraulic circuit and the temperature of said fluids, whether the container houses a liver or a kidney, and - the disposable part comprises: - a container that in turn comprises: - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container , and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container , - a container that in turn comprises: - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container , and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container , - a hydraulic circuit comprising one first tube that channels a perfusion fluid externally pumped to the container to the liver or kidney housed in a central portion inside the container , and a second tube that channels a vasodilator fluid, - a cannula that channels urine or bile outward from the liver or kidney, respectively, to an external reservoir through a first outlet opening in the container , <EOS>