Ronald Herrick donated a kidney to his brother on December 23, 1954. Ron was the first living organ donor, to his twin brother, Richard. Joseph Murray, the plastic surgeon who convinced them and their surgical teams that he had figured a way around the immune response, received the Nobel Prize in Medicine.
There's an interesting interview with Dr. Murray here:
Celebrating 50 Years of Successful Solid Organ Transplantationhttp://www.donatelifeny.org/transplant/murray.html