The question arose in the previous thread about Crassus: When the Roman government collected money, whether by taxation or donations from patricians, where did they put it? And how did they disburse it back out again to pay for Legions or grain? Presumably they dealt in actual physical coins; where they stored in a central location and literally counted out to whoever was going to buy grain or pay legionnaires, or how was it done?
In the Republic, the State Treasury was called the aerarium ("the place of bronze"). It was housed in the Temple of Saturn, namely in the vaults beneath the temple, in the Forum. There was also the publicum, which was a separate treasury "for the people," but I won't go into that here. The aerarium housed all the state revenues, and also some other important items (like the legionary eagles, certain laws inscribed on bronze, important documents, etc). In the Republic, the aerarium was administered by elected Quaestors, who were in turn under the direct supervision of the Senate. The two Consuls could withdraw from the aerarium at will, but usually at the discretion of the Senate. Then as now, most of the biggest fights in the Senate revolved around money.
The treasury held coins, of course, but we are fairly certain that the vast majority of state income did not come in as coin (see Hollander, Money in the Late Republic 2007). Those assets were variously disbursed in other ways of "parking" money, which I won't go into here. When the legions were to be paid, they were probably paid in coin, though much of their "pay" was disbursed in the form of equipment and supply fees. The payment of legions in all periods is a complicated and contentious topic. In order to pay the dozens of legions Octavian had under arms after the Battle of Actium, the amount of coins would have been truly staggering--an impossible number. When "purchasing" in large sums, the state almost certainly utilized measured bullion instead of coinage.
Augustus made some changes to the aerarium. He separated out a new fund, the fiscus, for Imperial business. It functioned basically as a private aerarium just for the Emperor, which the Senate could not manage. The regular aerarium continued to receive revenue from traditional sources, however, including "senatorial" province taxation. Control of the aerarium passed chaotically in the early Empire, from Quaestors to appointed Prefects to Praetors and back to Quaestors. Ultimately it wound up in the control of appointed prefects, who were former Praetors, and who were nominally under Senatorial management.
It's a big topic. If you have specific questions, I can (try to) answer them here.