First came the athletes, then the greasers. Then came the nonconformists, the teenagers and finally the baby boomers. The shoe manufacturer Converse has sold its brand of cool and whiff of rebellion to generations of Americans.

Now Converse, the century-old American footwear maker, says some of the core elements of its widely recognizable Chuck Taylor sneakers — black stripes and a rubber toe topper — are being co-opted by Walmart, Kmart, Skechers and others. And it is taking them to court, accusing 31 companies of trademark infringement in 22 separate lawsuits filed Tuesday in United States District Court in Brooklyn.

But while Converse, which has been owned by Nike since 2003, is suing for monetary damages, its main priority is getting impostors off the shelves. To that end, the company is pursuing a separate complaint with the International Trade Commission, which has the power to stop any shoes considered to be counterfeit from entering the country.

“The goal really is to stop this action,” said Jim Calhoun, the Converse chief executive. “I think we’re quite fortunate here to be in the possession of what we would consider to be an American icon.”