1) It is permitted to transplant or graft an organ from one place of a person�s body to another, so long as one is careful to ascertain that the benefits of this operation outweigh any harm that may result from it, and on the condition that this is done to replace something that has been lost, or to restore its appearance or regular function, or to correct some fault or disfigurement which is causing physical or psychological distress.
2) It is permitted to transplant an organ from one person�s body to another, if it is an organ that can regenerate itself, such as skin and blood, provided that the donor is mature and fully understands what he is doing, and that all the other relevant shariah conditions are met.
3) It is permitted to use part of an organ that has been removed from the body because of illness to benefit another person, such as using the cornea of an eye removed because of illness.
4) It is unlawful (haram) to transplant or use an organ on which life depends, such as taking a heart from a living person to transplant into another person.
5) It is unlawful (haram) to take an organ from a living person when doing so could impair an essential and vital function in his body, even though his life itself may not be under threat, such as removing the corneas of both eyes. However, removing organs which will lead to only partial impairment is a matter which is still under scholarly discussion, as is mentioned in resolution 8.
6) It is permitted to transplant an organ from a dead person to a living person when his life depends on receiving that organ, or when vital functions of his body are otherwise impaired, on the condition that permission is given either by the person before his death or by his heirs, or by the leader of the Muslims in cases where the dead person�s identity is unknown or he has no heirs.
7) Care should be taken to ensure that in all of the above situations where transplantation is permitted, no buying or selling of organs is involved. It is not permitted to trade in human organs under any circumstance. However, the question of whether the beneficiary may spend money to obtain an organ he needs, or to show his appreciation, is a matter which is still under the research of the scholars.

Anything other than the scenarios described above is still subject to scholarly debate, and requires further detailed research in the light of medical research and shariah rulings. And Allah knows best. (See: Qararat wa tawsiyat majma� al-fiqh al-Islami, P. 59-60).
The Islamic Fiqh academy of India also discussed this issue, and one of their members who is recognised as a renowned scholar Shaykh Ubaid Allah al-As�adi (Allah preserve him) compiled a very extensive article. The following are summarized points of some of the conditions mentioned in his article (that have not been mentioned in the above resolutions):
1) There should be no other reasonable alternative available in the view of experienced medical experts
2) All efforts should be made to avoid transplanting of a non-Muslim�s organ into a Muslim and vice versa. If this is not possible, only then the organ of a non-Muslim may be used.
3) Only the organs of a dead person should be used. However, if that is not possible, then only will it be permissible to use the organ of a living person.