In a Boy Named Sue, Johnny Cash sang of a parent who gives his son a name he knows will cause problems. But the wise father also knows that the painful challenges that go with the name will toughen the kid to survive in a mean world.

She isn't a girl named Rocky, but Johnny's eldest daughter, Rosanne, carries a surname with problems of its own. "Nobody likes the kids of famous people," she says matter-of-factly. "It's particularly hard if you go into the profession where the parent has been very successful. But if that's where your talent lies, it's dumb not to pursue it. Doctors' children become doctors. It shouldn't be all that strange that Johnny Cash's child likes to sing."

In any case, most country music critics agree that Rosanne could sell records even if her last name were Roseannadanna. At age 27, she has scored three No. 1 hit singles on the country charts in the past two years—Seven Year Ache, Blue Moon With Heartache and My Baby Thinks He's a Train—on her own. Her new album, Somewhere in the Stars, hit the country Top 10 after a month, and a single, Ain't No Money, got there too.

If she's aware of the burdens her name brings, Rosanne doesn't pretend that her Cash legacy has exactly hurt her career. "The media were naturally more interested in me because of my name—but they also expected a lot," she says. "I think the advantages and disadvantages about cancel each other out."

Rosanne was born in Memphis to Johnny and his first wife of 13 years, Vivian. Not long afterward, her dad became a part of the Million Dollar Quartet with Jerry Lee Lewis, Carl Perkins and Elvis. "He wasn't around much," she recalls. "He was so driven and ambitious." Johnny for a time had trouble with both pills and booze, and he split with Vivian when Rosanne was 12. "My dad had his own personal problems," she says. "I was a girl, and so naturally I went to live with my mother."

Rosanne spent her teenage years growing up with her mom, stepfather Dick Distin, a private investigator, and three sisters in Ventura, Calif. "We were spoiled," Rosanne admits, "but weren't brats. If I wrecked the car, I knew I had to get a job to pay for it."

At 18, Rosanne decided to pass up college and become a backup singer for her father, who had married June Carter in 1968. For three years she traveled the world with them. "I guess I got a little too cocky," she says. Humility came when she went to live in London, supposedly to study drama, but found she lacked the nerve even to apply to the Royal Academy of Dramatic Art. Instead, she worked as a secretary in the London office of CBS Records—her father's label.

She returned to the U.S. to attend college at Vanderbilt in Nashville and was served another slice of humble pie when she couldn't squeeze into the campus drama circle. After a year she dropped out and moved to Hollywood and lived alone while going to acting school. "I was robbed twice," she says. "Guys would follow me home from the supermarket. I'm amazed I wasn't killed." Her six months at the Lee Strasberg Theatre Institute, she says, "taught me how to deal with emotional realities and express them," though she's never acted professionally.

Soon after, at a party at Waylon Jennings' house in Nashville, she met Rodney Crowell, a singer, guitarist and composer of such hits as 'Til I Gain Control Again. "When I got the chance to record," she says, "I asked him to be my producer." Their musical relationship blossomed into romance, and they were married in 1979.

They have learned to settle the spats she admits they used to have in the studio. "Now it's very romantic—we don't take the saxophone track to bed with us." They live in a spacious but homey cabin nestled in 11 acres of woods in the Tennessee countryside outside Nashville. Their only bow to their California past is a hot tub. They have three children: Hannah, 6, a daughter from Rodney's previous marriage; Caitlyn, 2; and Chelsea, born last January.

Rosanne likes living near Nashville, she says, because it's near her clan. "He loves the kids and us, but he's on the road a lot and lives his own life just like us," she says of her dad, whose spread is only 45 minutes away. The family's other second-generation singer, stepsister Carlene Carter, June's daughter by her first marriage, lives in London with her husband, British rocker Nick Lowe. Notes Rosanne: "We aren't exactly in each other's laps. I'm just the sister who's always pregnant or shlepping my kids around in a station wagon."

Having conquered the challenges of living down and up to the Cash name, what would she think of a third generation of country-singing Cashes, including the Crowell kids? "I think it would be great," Rosanne says, pausing for effect, "if they were doctors or archaeologists or astronauts."