The case was brought to the Supreme Court by The Slants after their trademark was refused because the name is disparaging to Asian-Americans.
The Asian-American band acknowledges the name may offend, but say they chose it to "reappropriate" its meaning.
The ruling could impact the high-profile case of the American football team, the Washington Redskins.
In 2014 the US Patent and Trademark Office cancelled the team's six trademarks, including the lucrative team logo, after years of complaints from Native American groups.
