The lawsuit, filed Wednesday in San Francisco Superior Court, described attacks that occurred in several states from January 2018 through June 2019, according to media reports.
The lawsuit claims the San Francisco company didn't respond properly when the women reported the incidents; allowed drivers who had sexual-assault complaints lodged against them to continue to drive for the service; does not proactively report incidents to the police, and didn't respond to detectives who were handling cases, media reports say.
Lyft also does not have an anti-sexual-harassment training program for its drivers, according to the lawsuit.
The suit called on the company to take actions including making more thorough background checks, installing video cameras in cars for monitoring, and placing a panic button in the app, reports said.
In a statement to CNN, Mary Winfield, Lyft's head of trust and safety, called the allegation descriptions "terrifying."
"As a platform committed to providing safe transportation, we hold ourselves to a higher standard by designing products and policies to keep out bad actors, make riders and drivers feel safe, and react quickly if and when an incident does occur," Winfield told CNN.
"Our commitment is stronger than ever, as we dedicate more resources in our continued effort to ensure our riders and drivers have the safest possible experience."