The riders are also seeking compensation from Uber due to the alleged rapes the women suffered.
An Uber spokesperson said: "These allegations are important to us and we take them very seriously".
The lawsuit hits Uber months after the company over the treatment of women in its corporate offices.
"Had Uber not sacrificed passenger safety for the sake of profit and expansion, and actually cared about who it was employing to drive its vehicles, rather than being preoccupied with racing to control its share of the taxi market, at the expense of existing taxi companies and consumers, Plaintiffs herein and proposed Class members would not have been harmed", the complaint says.
"Instead, over the last seven years, Uber has done everything possible to continue using low-priced, woefully inadequate background checks on drivers and has failed to monitor drivers for any violent or inappropriate conduct after they are hired". Nothing meaningful has been done to make rides safer for passengers - especially women.
The lawsuit describes all the access Uber provides for drivers to commit sexual assault.
We've reached out to Uber and will update when we hear back. In June, CEO Travis Kalanick resigned from Uber, following criticism that he helped foster a culture of sexism and bullying at the company. Buttoning up these legal issues, and cutting its financial losses, will become increasingly imperative as the company prepares to go public in the next 18 to 36 months.
Uber has always been under fire for its background check methods. It's unclear whether Uber received complaints specific to these drivers.
It says the firm has resisted changes, such as more stringent background checks, that would improve passenger safety.
The document reads that many US states have strict monitoring of taxi drivers, and some even require cameras in vehicles, but Uber does not have comparable regulations.
Concerns about Uber's screening procedures for drivers have dogged the company for a number of years. In fact, the state found that 51 of those drivers who were rejected were sex offenders and 352 of them had been accused of incidents related to "sex, abuse and exploitation".
Uber is facing a class-action lawsuit over sexual assaults by drivers.
"To skirt state and local regulatory scrutiny, Uber labels itself a "technology platform" company rather than a "transportation" company", the complaint reads.
Uber regulations differ for Canadian cities. It is time for Uber to "Do the right thing. Period." That includes divulging how many reports of sexual assault Uber receives and what it does with those reports.
Alleged incidents have occurred in many countries where Uber operates, and have resulted in several high-level employees departing from the company.