"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".
The women from Florida and California are not identified in the federal lawsuit filed Tuesday in San Francisco.
The proposed class-action also lists a number of stories about other women accosted by Uber drivers, some of whom shared these details via Twitter under the hashtag #metoo, which is created to allow women to voice their stories of sexual assault and harassment.
Two anonymous plaintiffs who were assaulted by Uber drivers filed the lawsuit.
States have tried to force Uber's hand on the issue to no avail.
Reviews of Uber driver applications by two U.S. states, Maryland and MA, have led to rejections of thousands more applications than under Uber's own system. Nothing meaningful has been done to make rides safer for passengers - especially women.
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"In our complaint filed today against Uber, the allegations show that Uber's model of profits over safety has caused countless numbers of female riders to pay the price for Uber's shameless failure to act", the plaintiffs attorney said in a statement.
USA Today reported that the women are alleging that Uber's lax screening of its drivers has subjected thousands of female passengers to a range of sexual harassment or sex assault. The complaint cites accounts of assault by Uber drivers shared as part of the #MeToo movement; failures of background checks, including the registration as an Uber driver of the truck driver in the terrorist attack in NY on October 31; the debate over whether Uber drivers are employees of Uber; and misleading advertising from Uber about "safe rides".
The women are requesting damages for their own alleged assaults along with an injunction against the company that would require it to conduct more rigorous background checks on its drivers and change other company practices.
The plaintiffs in the case call for more transparency from Uber's new CEO Dara Khosrowshahi, who recently overhauled the company's values to include: "We do the right thing".
The lawsuit describes all the access Uber provides for drivers to commit sexual assault. Uber says the groups have helped it train 150 customer service agents for a team to deal with sexual assault reports.