It has been discovered that there are at least two significant antigens present on the cells of animal species such as pigs that elicit an immune or inflammatory response immediately upon implantation into humans or contact with human serum. The first is an .alpha.-galactosyl (Gal) epitope, for example, Gal.alpha.(1->3)Gal.beta.(1->4)GlcNac (linear B type 2) or Gal.alpha.(1->3)Gal.beta.(1->4)Glc (linear B type 6). The second is an N-glycolylneuraminic acid (NeuGc) structure. By eliminating these epitopes, preferably by genetically engineering the animal so that the epitope is either not produced or is greatly reduced, or by chemical or enzymatic treatment of the animal's cells to remove the epitopes, it is possible to produce organs, tissues and cells suitable for xenotransplantation into humans. Cells can be rendered even more compatible by genetically engineering the animal to express a human complement regulator protein (inhibitor), such as CD59, on its cells, or to express an excess of a pig complement regulatory protein.