The Vatican has never definitely stated whether the Shroud is authentically the burial cloth of Christ. Which is actually a method it takes with many relics of this nature. That being said, in the past many Popes have taken its authenticity for granted. It has, however, been approved to be part of a series of devotions known as "The Holy Face of Jesus" and is included in the repertoire of relics including Veronica's veil and the Sudarium of Oviedo.
The image on the cloth that King Abgar saw and was converted by is known as the Image of Edessa, and in Eastern Christianity as the Mandylion. In the film the makers take for granted that the image of Edessa is in fact the Shroud of Turin folded up so as to only show the face. There is historical evidence to support this claim, however there is no current consesus by historians.
About a decade after the film came out a carbon dating was done on the shroud. It gave a date of around 1260-1390 AD. However the results have been seriously questioned in the years that followed because of flawed sampling, issues relating to the fire that damaged it, and the samples being taken from parts of the cloth that may have been rewoven by French artisans in the renneisance.
In 1983 the Savoy family gifted the Shroud to the Holy See.