For patients with severe kidney or liver failure the best solution is currently organ transplantation. However, not all patients are eligible for transplantation and due to limited organ availability, most patients are currently treated with therapies using artificial kidney and artificial liver devices. These therapies, despite their relative success in preserving the patients' life, have important limitations since they can only replace part of the natural kidney or liver functions. As blood detoxification (and other functions) in these highly perfused organs is achieved by specialized cells, it seems relevant to review the approaches leading to bioengineered organs fulfilling most of the native organ functions. There, the culture of cells of specific phenotypes on adapted scaffolds that can be perfused takes place. In this review paper, first the functions of kidney and liver organs are briefly described. Then artificial kidney/liver devices, bioartificial kidney devices, and bioartificial liver devices are focused on, as well as biohybrid constructs obtained by decellularization and recellularization of animal organs. For all organs, a thorough overview of the literature is given and the perspectives for their application in the clinic are discussed.