In the early 1200s the city of Siena was growing quickly, and among other concerns was the difficulty of providing water to the city. When wells could not keep up with demand, some unknown engineer came up with a cunning solution: dig the wells sideways, and keep digging until you get a stream.
This rather odd solution works because of the hydrology of the surrounding hills. The tunnels would travel gently uphill, staying above the local aquiclude of clay; as they traveled through the permeable limestone they would collect trickles of water from the surrounding aquifer, until they collected enough to run the fountains and fill the wells of Siena.
These underground aqueducts were collectively known as bottini -- from the Italian botte, meaning barrel -- because they were usually finished with vaulted ceilings. They proved quite popular, in part because the soft rock made for quick and easy digging, but mostly because the need for water encouraged constant expansion. The tunnels were excavated easily using a pick, and were often finished with brickwork too support the walls and ceiling. Because connecting tunnels is difficult without advanced technology, the bottini were often twisty and connected with odd joints and shafts. They were occasionally punctuated with 'eyes', vertical shafts (now capped) that allowed diggers to determine where exactly they were, and perhaps also to allow for easier removal of the excavated rock.
The current network is said to extend for over 25 kilometers (15 miles), with main branches travelling outside the city and under surrounding towns. These were a major municipal project (first funded by the wool guild, and then later by the city), originally constructed to maintain a rather famous public fountain, the famous Fontebranda. This portion of the project lasted from at least 1226 to 1267, with various starts and stops. In 1343 a larger construction, feeding the Fonte Gaia was undertaken.
It perhaps worth noting that these fountains were, and still are, what we might today call a pool or a cistern. They were not pressurized, and did not spray or spout. While they were often quite decorative, they were also highly functional, providing water to households, livestock, and for washing clothes. Maintaining these sources of water was a matter of great importance to the citizens and local industries.
There are currently at least four networks of bottini, depending on your definition. The Fonte Gaia network is the largest, and only it and the Fontebranda network are particularly impressive; none of the others stretch for kilometers or have multiple branches. The Fonte Gaia and Fontebranda networks were connected in 1825, a connection that was not made earlier because the Fonte Gaia is above the Fontebranda system, and without an effective pumping system it would simply drain into the lower network.
While there are no certain statistics to be found -- in English, at least -- it appears that at their height these systems supplied water to more than fifty public fountains and wells. While at no point during the Middle Ages or Renaissance did Siena have as much water as it wanted, the bottini allowed it to grow and prosper more than it could have otherwise.
The bottini were, and still are, difficult to maintain. They need upkeep not just to repair brickwork and cave-ins, but also clearing out silt, tree roots, and calcium carbonate deposits. Historically they have also needed to be guarded from those who would use them for thievery or invading the city (as was attempted by an invading pope in 1526).
In the 1800s the system was modernized to the point where many households used the bottini as a water source for their personal wells. Branches were dug under the houses and plugged with stone plates; the homeowner could then pay to have holes drilled in the plate to release water. These holes were called dadi, and one could pay for as many holes as one wanted (apparently, half a hole was also an option), with one hole passing 400 liters a day.
At this time, tiled street signs and house addresses were inlaid in the walls so that the dadi
could be properly managed. Since that time there has been comparatively little development to the bottini, although they still functioning today. You can tour them for a fee, although you will have to book in advance.
In other parts of the world this sort of construction may be called a qanat.
Sources and further reading:
A good overview.
One more map.
More exhaustive information.