The first successful human transplantation took place in 1954 in Boston at the Brigham Hospital. This was between two identical twins and it was highly successful. By transplanting between identical twins, the immunological barriers were overcome and did not come into play.
Obviously this was an ethically challenging operation; taking one organ from a perfectly healthy individual and giving it to another with an uncertain future. Utlimately it was the correct decision, in that the organ was still working 8 years later.