The great sewers of the Roman Empire are one of the oddities of Roman engineering in that they weren't exactly built to be sewers in the first place -- as immense and complex as they were, they weren't so much invented as they just sort of happened. The Cloaca Maxima (or Biggest Sewer if you want to translate it directly) was originally just a channel built to drain some local marshes. Digging commenced around 600 B.C., and over the next 700 hundred years, more and more waterways were added. Since more channels were dug whenever it was deemed necessary, it's hard to tell when the Cloaca Maxima stopped being a drainage ditch and became a proper sewer. Primitive though it was initially, the Cloaca Maxima spread like a weed, stretching its roots deeper and deeper into the city as it grew.
Unfortunately, because the Cloaca Maxima drained directly into the Tiber, the river became absolutely swollen with human waste. That's certainly not an ideal situation, but with their aqueducts, the Romans didn't need to use the Tiber for drinking or washing. They even had a goddess to watch over their system -- Cloacina, the Venus of the Sewer.
Perhaps the most important and brilliant innovation of the Roman sewer system is the fact that it was (eventually) covered, cutting down on disease, smells and unpleasant sights. Any civilization can dig a ditch to go to the bathroom in, but it takes some impressive engineering to monitor and maintain a sewer system so complex that Pliny the Elder even declared it more stupendous than the Pyramids as a monument to human achievement.