The Museum of the Shroud is a focused Turin museum devoted to the history and study of the famous relic kept in the city. It covers photography, scientific investigation, religious devotion, and the long argument over what the cloth is and how it should be understood.
The actual Shroud is not normally on display, so come for context rather than spectacle. It is a useful stop before or after visiting the cathedral area, especially if you want to understand why the object still matters so strongly in Turin.