A new study by Italian scientsts doesn't really answer that question, but they make a convincing case that it isn't a medieval forgery.
"The double image (front and back) of a scourged and crucified man, barely visible on the linen cloth of the Shroud of Turin, has many physical and chemical characteristics that are so particular that the staining ... is impossible to obtain in a laboratory," concluded experts from Italy's National Agency for New Technologies, Energy and Sustainable Development.
The scientists set out to "identify the physical and chemical processes capable of generating a colour similar to that of the image on the Shroud." They concluded that the exact shade, texture and depth of the imprints on the cloth could only be produced with the aid of ultraviolet lasers - technology that was clearly not available in medieval times.
The scientists used extremely brief pulses of ultraviolet light to replicate the kind of marks found on the burial cloth.
They concluded that the iconic image of the bearded man must therefore have been created by "some form of electromagnetic energy (such as a flash of light at short wavelength)." Although they stopped short of offering a non-scientific explanation for the phenomenon, their findings will be embraced by those who believe that the marks on the shroud were miraculously created at the moment of Christ's Resurrection.
"We are not at the conclusion, we are composing pieces of a fascinating and complex scientific puzzle," the team wrote in their report.
A recent History Channel production, "The Face of Jesus" found some remarkable elements in the Shroud that proves it wasn't painted on or faked in a way that was possible during the time period it surfaced (mid 1300's). When placed under a sophisticated imaging device, the markings are shown to be three dimensional - impossible if a hoaxer were to paint the image or use any other technique known at the time to put it on the Shroud.
"For those who believe no explanation is necessary, for those who don't none will suffice."(Attributed to Joseph Dunninger.)