The paper deals with the concept of the identity of living organisms, a concept used up until now very ambiguously. The discussion rests on the combination of two concepts, one proposed by Munzer (1993) and another derived from the considerations of Riedl (1975). The first is the proposal that the identity of living organisms depends on the properties of their elementary constituents, such as cells and tissues, and that these properties, in turn, depend on those of their DNA and RNA. It follows that the identity of a living organism remains constant or changes during life according to whether its DNA and RNA content also remains constant or changes. The second is the consideration that, during duplication of a cell population, the informational content of the population does not increase if the duplicated cells are identical (increase only of redundant DNA). On the other hand the informational content of the cell population increases if the duplicated cells are the result of a variation-selection process (increase of essential DNA). The changes of DNA and RNA content, occurring in the germinal cells during phylogenesis and in the somatic cells of the evolutionary systems during ontogenesis, lead, therefore, to the generation of new identities. Living organisms are suggested to reflect two types of identity, that of the deterministic and that of the evolutionary systems. Since the informational content of the deterministic systems (the essential DNA content) remains approximately constant during life, their identity also remains constant. The changes in the number of elementary constituents and cell volumes during the processes of hypertrophy and atrophy are accompanied only by changes in the amount of DNA (the redundant DNA). On the other hand the informational content of the evolutionary systems (the essential DNA), such as the brain-mind system, the immunological system and some receptor systems, undergo a marked increase during the ontogenic development: this leads to changes of identity of these systems. For example, in the immunological system the process of mutation and recombination of the DNA of the immunological cells leads to the generation of new proteins in the amount about 10,000 times larger than that produced through the decodification of the genome. Also the construction of the neural network, and of a number of synapses much larger than that of the neuronal cells, requires the generation of an amount of new information much larger than that contained in the genome. In short, the attribution of a double identity to living organisms reflects the simultaneous presence of systems developing either within strictly programmed limits or without programs and limits, say as closed or open projects. The difference between the two types of systems explains the different effects in the case of the transplants. The identity of the recipient of transplants is not altered in the case of transplants of a deterministic system but is so in case of transplants of evolutionary systems. There is now a widespread fear of the possibility of human cloning. It is argued that this fear is unjustified because a cloning process can never succeed in duplicating those parts which are essential for the characters of humans, namely those concerned with the properties of the evolutionary systems.