The fast-rising conductor Eun Sun Kim was named the next music director of the San Francisco Opera on Thursday, which will make her the first woman to hold the post at an American opera company of major size and stature.

When asked in an interview about the pathbreaking nature of her appointment, Ms. Kim, 39, who was born in Seoul, South Korea, said that it made her think of how her grandmother, a pioneering doctor who was born in 1912, was described for years as a “female doctor” — but lived to see the day when women who practiced medicine were called, simply, “doctors.”

“So I’m grateful to be the first ‘female music director,’” Ms. Kim said. “But I also look forward to a future where the next generation will be called just ‘conductor.’”

Ms. Kim, whose conducting career has boomed in recent years, regularly conducts in Europe, including at the Staatsoper in Berlin, where Daniel Barenboim invited her to make her debut in 2015. She made her American debut in 2017 under the most trying circumstances: conducting “La Traviata” at the Houston Grand Opera in an improvised theater in a convention center, shortly after Hurricane Harvey flooded the company’s regular home.