The replacement of diseased or damaged organs by transplantation became a reality in 1954, when a kidney was transplanted from one monozygotic twin to another. Barring technical difficulties, a graft between genetically identical (syngeneic) individuals is readily accepted and is termed an isograft. An allograft is a transplant between allogeneic (genetically not-identical) members of a given species, whereas a xenograft crosses species lines. Allografts and xenografts provoke a strong immunological response which, unless suppressed, leads to rejection and graft loss. Immunological processes involved in rejection include humoral (B-cell) and cellular (T-cell) components that are both antigen specific and non-specific. 45 years ago Peter Medawar and his colleagues established that rejection has an immunological basis. From studies of skin grafts between animals, they demonstrated that transplantation immunity was actively acquired, systemically propagated and highly specific. Previous exposure to donor antigens was shown to lead to the sensitization of the recipient and resulted in a markedly accelerated rejection upon re-exposure to the same antigens. Also at this time, the possibility of transplantation tolerance, or specific allograft unresponsiveness was first described in an animal bone marrow chimera model. The greatest challenge in clinical transplantation today is to develop in the organ recipient a state of immunologic non-responsiveness to the organ while preserving the ability for immune surveillance and function necessary for life.