Transplants that do not require the death of the donor, kidney transplants etc., are certainly permitted as the donor is capable of surviving with only one kidney, whereas the donee will not be able to live without the transplant. That is why immediate family are often encouraged to be donors because statistics prove that their organs have a better chance for successful transplantation than others.
In theory organ donations are permissible after death as defined by Jewish Law. However, the whole subject is fraught with problems since the moment of death according to Jewish Law is not the same as according to civil law. Subsequently, the greatest authorities in Jewish Law of our generation are hesitant to give a general answer.
Rabbi Moshe Feinstein ruled that one cannot use hearts for transplants because when heart transplants were first introduced it was imperative to use the heart of the donor before the donor could be classified as dead according to Jewish Law. Today, with the advancement of technology such a ruling may not be applicable in every case, and therefore each case should be judged by a Torah scholar who is also an expert in this subject. This is the case for all potential donors and donees for transplants. The people involved need to discuss their case with a Torah scholar who is an expert in the field because each case must be judged individually.
