Some histories of organ transplantation

I can't vouch for any of these...most are un-refereed internet pages...some starting with events reported from quite long ago, regarding skin and bones, for example.

From organdonor.gov: Timeline of Historical Events Significant Milestones in Organ Donation and Transplantation

http://www.organtransplants.org/understanding/history/


http://optn.transplant.hrsa.gov/learn/about-transplantation/history/

http://www.donatelifeny.org/all-about-transplantation/organ-transplant-history/

https://www.unos.org/transplantation/history/

http://www.mtf.org/news_history_of_transplantation.html


Here's a journal article, whose history begins with kidneys...
Organ transplantation: historical perspective and current practice

C. J. E. Watson1,* and J. H. Dark, 

Br. J. Anaesth. (2012) 108 (suppl 1): i29-i42.

doi: 10.1093/bja/aer384 

"A brief history of transplantation
Kidney transplantation
Since Jaboulay and Carrel developed the techniques
required to perform vascular anastomoses at the turn of
the last century, there has been a desire to treat organ
failure by transplantation. Jaboulay was the first to
attempt this in 1906, treating two patients with renal
failure by transplanting a goat kidney into one and a pig
kidney into the other; in both cases, he joined the renal
vessels to the brachial vessels.1 Both transplants failed
and both patients died. At that time, there was no alternative
to death if renal failure developed, and it would be
another 38 yr before the first haemodialysis machine was
invented. The first use of a human kidney for transplantation
followed in 1936 when Yu Yu Voronoy, a Ukrainian
surgeon working in Kiev, performed the first in a series of
six transplants to treat patients dying from acute renal
failure secondary to mercury poisoning, ingested by its
victims in an attempt to commit suicide. All the transplants
failed, in large part because of a failure to appreciate the
deleterious effect of warm ischaemia; the first kidney was
retrieved 6 h after the donor died.
One limitation to transplantation then, as now, was the
lack of suitable donor organs. The initial pioneers had used
animal organs or organs from long deceased humans. In
the 1950s, there came a realization of the need to avoid
excessive ischaemic injury and kidneys from live donors
began to be used. Some of these were from the relatives of
the recipient; others were unrelated patients having a good
kidney removed for other reasons. The surgical technique
also needed refinement; while a kidney based on the thigh
or arm vessels might be technically straightforward, and possibly
adequate for the short-term treatment of acute renal
failure, it was not a realistic solution for the long term.
That solution came from France in 1951 and involved
placing the kidney extraperitoneally in an iliac fossa, where
the external iliac vessels are easy to access and the
bladder is close by for anastomosis to the donor ureter;
this is the technique still used today.
Having overcome the technical issues of vascular anastomosis
and placement of the kidney, there remained the
problem of the immune response. Medawar’s work during
and after the Second World War studying the rejection of
skin grafts had demonstrated the potency of the immune
system.2 At that time, attempts to control the immune
system using irradiation had proved either ineffectual or
lethal. The first successful transplant therefore came about
by avoiding an immune response altogether, which Joseph
Murray’s team achieved by performing a kidney transplant
between identical twins.3 There then followed a series of
identical twin transplants around the world, with the first in
the UK being performed in Edinburgh by Woodruff and
colleagues4 in 1960."