Marilyn and John McFarlane in Lighthearted Travel say it well:
Over the years various scientific tests have been performed on the fabric. Every finding, on either side, is challenged. Nothing guarantees its authenticity–and nothing explains the imprint of the image. It remains shrouded in mystery. My eyes still see only an ancient, stained cloth, but the enlargements and displays in the museum point it out clearly. If it’s a fake, it’s impressive. Even more impressive is its effect on the millions who hold it sacred.