The complaint alleges the women were attacked in their rides and that Uber has known about sexual misconduct by some of the drivers since 2014.
The complaint was filed in San Francisco, where the company is based.
"Uber's whole business model is predicated on giving people a safe ride home, but rider safety was never their concern - growth was, at the expense of their passengers' safety," said attorney Adam Slater, whose firm represents about 550 plaintiffs. "While the company has acknowledged this crisis of sexual assault in recent years, its actual response has been slow and inadequate, with horrific consequences."
Uber said it could not comment on the specific pending litigation, but noted that it has rolled out a number of features designed to improve rider safety.
"Sexual assault is a horrific crime and we take every single report seriously," an Uber spokesperson said. "There is nothing more important than safety, which is why Uber has built new safety features, established survivor-centric policies, and been more transparent about serious incidents."
The newer safety features include: the ability for riders to call or text 911 through the Uber app; RideCheck to monitor if a trip has gone off-course or if perhaps a crash has occurred; GPS tracking; and background checks on drivers.
The company says last year it also started sharing information with its rideshare rival Lyft about drivers who have been deactivated from the platform for safety issues.