The paper presents the problems connected with the modification of animal tissues through transgenesis. Special attention has been given to the xenogenic heart transplant between pigs and human beings. The heart of pigs is a highly discordant graft both for primates and human, which is hyperacutely rejected already several minutes after operation. Graft rejection is mainly due to the presence of natural xenoreactive antibodies which recognize the pigs' tissue epitop Gala1-3Gal (known as the Gal antigene), as well as to the quick activation of the complement in the recipient. In 1995 transgenic pigs with human genes responsible for complement CD55 (DAF - decay accelerating factor) and CD59 factor (membrane inhibitor of reactive lysis) were obtained. Organs of such animals, grafted into baboons, were resistant to complement-related rejection. The possibilities of modifying animal tissues by way of genetic manipulation are still limited and total 'humanzation' of pigs heart as regards the histocompatibility antigens is not yet feasible. For this reason the fulfilment of the idea of xenografting between pigs and human beings is still remote.