The lack of organs for transplantation is an important problem in medicine today. The growth of organs in chimeric animals may be the solution of this. The proposed technology is the interspecific blastocyst complementation method in combination with genomic editing for obtaining “free niches” and pluripotent stem cell production methods. The CRISPR/Cas9 method allows the so-called “free niches” to be obtained for blastocyst complementation. The technologies of producing induced pluripotent stem cells give us the opportunity to obtain human donor cells capable of populating a “free niche”. Taken together, these technologies allow interspecific blastocyst complementation between humans and other animals, which makes it possible in the future to grow human organs for transplantations inside chimeric animals. However, in practice, in order to achieve successful interspecific blastocyst complementation, it is necessary to solve a number of problems: to improve methods for producing “chimeric competent” cells, to overcome specific interspecific barriers, to select compatible cell developmental stages for injection and the corresponding developmental stage of the host embryo, to prevent apoptosis of donor cells and to achieve effective proliferation of the human donor cells in the host animal. Also, it is very important to analyze the ethical aspects related to developing technologies of chimeric organisms with the participation of human cells. Today, many researchers are trying to solve these problems and also to establish new approaches in the creation of interspecific chimeric organisms in order to grow human organs for transplantation. In the present review we described the historical stages of the development of the blastocyst complementation method, examined in detail the technologies that underlie modern blastocyst complementation, and analyzed current progress that gives us the possibility to grow human organs in chimeric animals. We also considered the barriers and issues preventing the successful implementation of interspecific blastocyst complementation in practice, and discussed the further development of this method.