The 'Roman Well' is located in the southwest part of the Upper Town within Kalemegdan fortress, though in truth the well really isn't Roman, at all.
More likely is that it was actually built in the 18th century by the Austrians and that the name could be connected to their ambition to be considered as the inheritors of the great Roman Empire, or as the holders of the title of the Holy Roman Emperor, a confusing title which came after the actual Roman empire and referred to the ruler of the Germanic lands. (There are in fact real Roman wells in Belgrade. One is located below the courtyard of the Russian Embassy in Vracar, and one of the former tram turntable in Zvezdara, and a third in the square between Faculty of Philosophy and Captain Misa building also called Njegos Square.)
According to the official story, the well was built during the Austrian reconstruction of the Belgrade Fortress in order to provide water for the army, between 1717 and 1732. Unfortunately, the digging proved to be futile and that might have changed its purpose, the well serving as a dungeon from then on. (Or more specifically it would have been, in French terms an "oubliette," or "forgotten place," a hole where prisoners were thrown in and forgotten about.)
The well was examined in detail in 1940, at which point the research team determined that it doesn't connect to the river Sava, but was instead being filled with surface water from the Upper Town.
The notoriety of the 'Roman Well' comes from the many tales and legends of prisoners being thrown down the hole throughout the Belgrade history. The latest case was surprisingly recent when in 1954 a mentally ill man killed his wife and threw her body down the well.