The veil believed by some to have been used to wipe the face of Jesus, known as the Veil of Veronica, is a relic that has a long and complex history. There are actually several claimants to be the "true" Veil of Veronica.
The most prominent claimant, and the one most often associated with the Vatican, is housed in Saint Peter's Basilica in Vatican City. However, it's important to note the following:
* It is not on permanent public display. The relic is rarely exhibited, and when it is, it is shown from a distance.
* Its authenticity is disputed. The Vatican makes no official pronouncements regarding its authenticity. The image on the veil is very faint, and its origins are unclear.
* There are other claimants. Several other cloths claim to be the Veil of Veronica, including one in Manoppello, Italy.
So, the answer is: A veil believed to be the Veil of Veronica *is* housed in Saint Peter's Basilica in Vatican City, but its authenticity is disputed, and it is rarely displayed.