Organ transplantation is a treatment method for various organ failures and other severe pathologies, applied in critical cases to save a patient's life. However, there is a severe shortage of donor organs world-wide, resulting in hundreds of thousands of patients being unable to receive the organs they need in time. One possible solution to this problem is xenotransplantation-transplanting organs from animals to humans. Experiments in xenotransplantation began in the mid-1960s, with primates considered as the first potential donors. However, for a number of reasons, they proved unsuitable as a source of organs, while pigs turned out to be the best donors. The development of modern genetic engineering and genome editing methods has led to a new perspective on these animals as a source of human organs. Various genetic modifications have significantly reduced the immune response of the recipient to the graft and improved survival. To date, several successful transplants of organs and tissues from wild-type and genetically modified pigs to humans have been carried out worldwide, and a few companies are developing specialized lines of animals for xenotransplantation. In this review, we provide a detailed overview of the history of xenotransplantation worldwide, as well as all the genetic modifications introduced into the genome of pigs the organs of which have been used successfully for human transplantation, the role of these genetic modifications, and the mechanisms by which they are introduced.