While there are variations in specific views, it is clear that most major religions of the world do in fact permit, allow and support transplantation and donation.
Episcopal
The Episcopal Church passed a resolution in 1982 that recognizes the life-giving benefits of organ, blood, and tissue donation. All Christians are encouraged to become organ, blood, and tissue donors "as part of their ministry to others in the name of Christ, who gave His life that we may have life in its fullness."
Islam
The religion of Islam strongly believes in the principle of saving human lives. According to A. Sachedina in his Transplantation Proceedings' article, Islamic Views on organ transplantation, "the majority of the Muslim scholars belonging to various schools of Islamic law have invoked the principle of priority of saving human life and have permitted the organ transplant as a necessity to procure that noble end."
Lutheran
In 1984, the Lutheran Church in America passed a resolution stating that donation contributes to the well-being of humanity and can be "an expression of sacrificial love for a neighbor in need." They call on "members to consider donating organs and to make any necessary family and legal arrangements, including the use of a signed donor card."
Romani or Roma People
The Romani or Roma people are a people of different ethnic groups without a formalized religion. They share common beliefs and tend to be opposed to donation. Their opposition is connected with their beliefs about the afterlife. Traditional belief contends that for one year after death, the soul retraces its steps. Thus, the body must remain intact because the soul maintains its physical shape.
Shinto
In Shinto, the dead body is considered to be impure and dangerous, and thus quite powerful. "In folk belief context, injuring a dead body is a serious crime…", according to E. Narnihira in his article, "Shinto Concept Concerning the Dead Hutnan Body." "To this day it is difficult to obtain consent from bereaved families for donation or dissection for medical education or pathological anatomy…the Japanese regard them all in the sense of injuring a dead body." Families are concerned that they not injure the itai - the relationship between the dead person and the bereaved people.
United Methodist
The United Methodist Church issued a policy statement in regards to donation. In it, they state that "The United Methodist Church recognizes the life-giving benefits of donation, and thereby encourages all Christians to become donors by signing and carrying cards or driver's licenses, attesting to their commitment of such organs upon their death, to those in need, as a part of their ministry to others in the name of Christ, who gave His life that we might have life in its fullness."