Modern interest in this famous relic was accentuated by the work of a team based in the USA which tended to demonstrate that the object came from the time of Christ. The development of accelerator mass spectrometry for radiocarbon dating at Oxford and elsewhere provided a tool which could be used to give an answer to the date of the original flax used and the consequent authenticity of the linen. An account of the period from the tests in 1978 until the final announcement in October 1988 is given.