The Shroud of Turin is a centuries old linen cloth that bears the image of a man who has suffered physical trauma in a manner consistent with crucifixion. Modern science has completed hundreds of thousands of hours of detailed study and intense research on the Shroud, making it the most studied artifact in human history.
It apparently appeared around the mid-1300 in Lirey, France and was transferred to Turin, Italy in 1578. Many people believe it is the burial shroud of Jesus. The one visible hand has a large, round wound, the side of the man has an upward gouge in it, and there are small punctures around the forehead and scalp, and scores of linear wounds on the torso and legs.
As much as that sounds like wounds the Bible describes Jesus suffering, there are some problems with this apparently blood-stained flax fabric. First, it carbon dates to around 1260 - 1390 with 95% confidence, almost that many years after Christ's crucifixion. Second, forensic tests describe the apparent blood as "tempera paint tinted with hematite (iron oxide). Third, Matthew, Mark and Luke all describe Jesus' burial cloths as strips of linen (multiple pieces). Fourth, a 1st Century burial shroud discovered in a tomb near Jerusalem in 2000 is composed of a simple "two-way weave" as opposed to the complex herringbone twill of the Turin Shroud. In the Shroud's favor, a textile expert has stated the 3:1 twill weaving pattern is consistent with 1st century Syrian designs, and a high resolution microprobe analysis of dirt particles on the Shroud are chemically identical to samples of limestone from ancient Jerusalem tombs.
I could go on and on. Crucifixion probably originated with the ancient Persians. There is evidence of crucifixion in the port of Athens in the 7th Century BC, and it was common during the reign of Alexander the Great (356 - 323 BC.). He crucified 2000 survivors of the siege of Tyre in 332 BC. The Romans adopted the custom from Carthage and used it for slaves, rebels, despised enemies and criminals. No one knows how many people Rome crucified, but they did approximately 6,000 of Spartacus' gladiators and followers along the 200 km road between Capua and Rome after one battle.
Crucifixion was abolished in the Roman Empire by Constantine in 337 AD, almost 1,000 years before the Shroud was made if the carbon dating is anywhere near accurate. It is possible the image on the Shroud is of a crucified man since crucifixion continued in other countries. We know Japan crucified people as recently as 1597, a missionary to Burma reported people being crucified there in 2000, and Saudi Arabia crucified a man as recently as March 2013.
In spite of all the evidence pro and con, it seems highly unlikely the "fine linen cloth" Jesus was buried in would have wound up 2200 miles (as the crow flies, at least twice that over land) from Jerusalem in France. The Catholic Church has not endorsed it, but they have kind of unofficially accepted it. I don't think we should do that.