How was the Shroud of Turin dated?

Answer

The cloth now known as the Shroud of Turin first appeared about 1355 at a little church in Liry, in north-central France. This places a latest date on the shroud.

In 1389 bishop Pierre D'Arcis wrote to the Avignon pope, Clement VII, that the shroud was being used as part of a faith-healing scam and spoke of a predecessor who conducted the investigation and uncovered the forger: "Eventually, after diligent inquiry and examination, he discovered the fraud and how the said cloth had been cunningly painted, the truth being attested by the artist who had painted it, to wit that it was a work of human skill and not miraculously wrought or bestowed." This places the shroud firmly in the fourteenth century.

Tests in 1988, by three laboratories (at Oxford, Zurich, and the University of Arizona) used accelerator mass spectrometry to date samples of the linen. The results were in close agreement and were given added credibility by the use of control samples of known dates. The resulting age span was circa 1250-1390 CE, which is entirely consistent with the correspondence from Bishop D'Arcis to Pope Clement VII.
After the carbon dating results became known, someone put out a false story that the tests were done on one of the patches from the 1532 fire, thus supposedly yielding a late date. A Russian scientist, Dmitrii Kuznetsov, claimed to have established experimentally that heat from a fire like that of 1532 could alter the radiocarbon date, but others could not replicate his alleged results and it turned out that his physics calculations had been plagiarised, complete with an error (Ian Wilson, The Blood and the Shroud). No credence can now be given to this falsified report, and so the carbon-dating results must stand.