Last weekend, at a tournament in the Cincinnati suburb of Mason, Coco Gauff beat Iga Świątek for the first time. It was one of those moments in tennis when the ground seemed to shift: Gauff had never taken a set from Świątek, the current world No. 1, in the seven previous times they’d met. It was the biggest win of Gauff’s young career—but it was in keeping with a high-summer revving of her already formidable game. In the hard-court tournaments held across North America which are essentially warmups for the U.S. Open, Gauff has been the imposing presence that the tennis world has been waiting for her to become—waiting avidly, for sure, but a little anxiously, too. As recently as early July, when she lost in the first round at Wimbledon, there was fretting that she wasn’t making quick enough progress. Then, three weeks ago, she won the Citi Open in Washington, D.C. A week later, in Montreal, she defeated the newest Wimbledon champion, Markéta Vondroušová. The victory over Świątek was a semifinal match; the following afternoon, she prevailed against the Czech Republic’s Karolína Muchová to capture the Cincinnati Open, the most notable tournament that Gauff has won so far. As the U.S. Open now gets under way, she finds herself someplace she has never been before: among the favorites to win a major championship.

Hard as it may be to believe, this is the fifth time that Gauff has had a spot in the U.S. Open’s main singles draw. She is still a teen-ager—she will turn twenty next March. She was only fifteen years old, and just months into her first season on the W.T.A. tour, when she caught the world’s attention by stunning Venus Williams in the first round at Wimbledon, in 2019. She left the court overwhelmed and teary; the Williams sisters had inspired her to take up tennis. People already knew that Gauff would be very, very good; she was a world No. 1 as a junior player. But it became clear that day that she had something else, too—that ineffable admixture of self-assurance, charisma, emotional vulnerability, and magnetism that could make her a pop-culture phenomenon. In America, but not only in America, she became Coco, full stop. And, in the absence of Serena, who retired from the tour last year, and Naomi—as in Osaka, who gave birth to a daughter in July, and has been away from the tour for long stretches dealing with injuries and mental-health issues—Coco has been the player whom many fans want most to see. Tennis, being mostly an individual sport, needs not just champions but stars. This may account, at least in part, for the anxiety there has been about Gauff. When would her game reach its full potential and match—and then, inevitably, raise—her popularity?

From the start, Gauff’s main coach has been her father, Corey. He brought her on tour relatively early, but also carefully. The hope, of course, was that she would be a champion, but also that she would be as comfortable with herself and her life as a tennis phenom can be. Last year, after reaching the final of the French Open (where she was overpowered by Świątek) and the quarterfinals of the U.S. Open, she broke into the Top Ten. She is currently No. 6 in the world. The last few steps to the summit of the game are the hardest. Given the point system that determines rankings, it is difficult to reach No. 1 without winning a major tournament. Teen-agers do win majors, even as the sport has become more physically demanding: four years ago, Bianca Andreescu won the U.S. Open as a nineteen-year-old; two years ago, Emma Raducanu won it at the age of eighteen. Both have endured injury after injury following their major triumphs and are currently ranked outside the Top 50.

Corey Gauff, who played basketball in college, had never been a serious tennis player, and for most of his daughter’s professional career he has shared coaching duties with an experienced tour-level coach. This summer, it has been Pere Riba, a onetime Spanish star who previously worked with the young Chinese comer Qinwen Zheng. Riba introduced a tactical change: he wanted Gauff to go bigger more often on her first serve—which has hit a hundred and twenty-eight miles an hour, as big as a serve gets in the women’s game—and, when the opportunity was there, to end points in a hurry. “I should be playing that aggressive style, focussing on my serve-plus-one,” Gauff told the Washington Post before the Citi Open, referring to the power-player strategy of landing a big serve, getting a weak return, and then pouncing on it, aiming to hit a winner and end the point.

Gauff’s strengths on offense have always been obvious: the first serve, the foot speed, the fearsome backhand she can angle crosscourt or drive down the line, the soft hands at the net, the court savvy, the fight. Her weakness—equally obvious, to her opponents—has been her forehand. Gauff employs a so-called Western grip on her forehand side, her palm beneath the racquet, her swing pattern long to muster pace. It’s a shot that creates a lot of topspin, which, in turn, creates a high, difficult bounce. When timed well and hit hard enough, it can generate a so-called heavy ball, one hard to return with force and control. But the Western forehand is not forgiving. It requires precise timing, exacting footwork, and fierce racquet-head acceleration through contact. Each one of these parts is vulnerable to pressure at tight moments in a match, and Gauff’s forehand has let her down on a lot of crucial points.

Riba has not changed Gauff’s grip. He has worked, instead, to instill in her a mind-set to end points more quickly—avoiding lengthy forehand-to-forehand rallies—and to ease the pressure that builds at pivotal moments by increasing her confidence. Corey Gauff has stepped back from his day-to-day coaching duties, and has not been travelling with Coco; another former player, Brad Gilbert, has joined Coco’s team as a consulting coach, to help in this effort, and to serve as a kind of parental figure in her coaching box. Gilbert, who’s sixty-two, is best known these days as an ESPN commentator, but he reached No. 4 in the world in 1990, mostly through stubborn defense and guile, and for eight years he coached Andre Agassi, helping him to become a more cunning, self-assured player and win a half-dozen major titles. Gauff sees Gilbert playing a similar role for her. “Just a lot of wisdom and how to play the points,” she said, of his influence, after she defeated Świątek in Cincinnati. “I think he’s one of the best tennis minds out there. . . . He’s just making me more serious.”

It’s a good guess that Gauff will win a major soon enough, and then win more. But it’s only a guess. How many times have talent and possibility been thwarted by injury, frayed desire, or sheer bad luck? In the meantime, Gauff is providing a special kind of enjoyment to attentive fans of hers: the pleasure—absorbing, fitful, at times remarkable—of watching so promising and ambitious and captivating a player become. “I’m still developing,” she said after the Świątek match, mindfully reminding us all. It has been, and remains, a special sort of unfolding, something to see.