Daniel Ek says his company is “not in the music space—we’re in the moment space.” Illustration by Harry Campbell; photograph: Eyevine / Redux

Daniel Ek, the C.E.O. of Spotify, is a rock star of the tech world, but he is not long on charisma. At thirty-one, he is pale, boyish, cerebral, and calm. Jantelagen, the Scandinavian code of humility and restraint, is strong in him. He doesn’t greet you with a firm handshake from behind an imposing desk; he doesn’t have a desk. He sprawls on a couch with his laptop, like a teen-ager doing homework. Or he wanders the company’s offices, which form an oval around the open core of a big building on Birger Jarlsgatan, in central Stockholm. The design encourages “random encounters,” which Ek once read was Steve Jobs’s plan in laying out Pixar’s offices.

Ek’s phlegmatic manner makes his unshakable, almost spiritual belief in Spotify burn all the more brightly. His vision, that Spotify is a force for good in the world of music, is almost Swedenborgian: salvation in the form of a fully licensed streaming-music service where you can find every record ever made. Spotify doesn’t sell music; it sells access to it. Instead of buying songs and albums, you pay a monthly subscription fee ($9.99), or get served an ad every few songs if you’re on the free tier. You can listen to anything on the service—the Beatles (as with iTunes, the surviving members are not rushing in) and Taylor Swift (who left the service in a flurry of publicity in early November) notwithstanding—and there is an astonishing amount of music. When Spotify launched, in October, 2008, in Sweden and a handful of other European countries, Ek’s dream seemed like the longest of long shots. Now Spotify is the Netflix of music sites. Mark Zuckerberg, Facebook’s founder, says, “Daniel just saw the opportunities of streaming music before anyone else.”

Spotify appeared nine years after Napster, the pioneering file-sharing service, which unleashed piracy on the record business and began the cataclysm that caused worldwide revenues to decline from a peak of twenty-seven billion dollars, in 1999, to fifteen billion, in 2013. The iTunes store, the industry’s attempt, in partnership with Apple, to build a digital record shop, opened in 2003 to sell downloads, but that didn’t alter the downward trajectory; indeed, by unbundling tracks from the album, so that buyers could cherry-pick their favorite songs, Apple arguably hastened the decline. Legal actions against individuals—thousands of people in the U.S. were sued for downloading music illegally—only alienated potential customers. As bad as the bloodbath was in the U.S., the situation was even worse in Sweden. Pelle Lidell, an executive with Universal Music Publishing in Stockholm, told me that by 2008 “we were an inch away from being buried, and Spotify single-handedly turned that around.”

Ek was one of the pirate band. Before starting the company, he had briefly been the C.E.O. of uTorrent, which made money in part by monetizing pirated music and movies on BitTorrent, a major file-sharing protocol. Later, the Napster co-founder Sean Parker, for years Public Enemy No. 1 to record-company executives, joined forces with Ek. Who would have imagined, as one label head put it recently, that “your enemy could become your friend”?

Spotify is now in fifty-eight countries. (Canada, its latest market, got the service at the end of September.) It has raised more than half a billion dollars from investors, including Goldman Sachs, to fund its expansion, and there are rumors of an I.P.O. in its future, to raise more. Spotify’s user base exceeds fifty million globally, with twelve and a half million paying subscribers. At the current rate of growth, that number could reach forty million subscribers by the end of the decade. To date, it has paid out more than two billion dollars to the record labels, publishers, distributors, and artists who own the rights to the songs. “I’m very bullish on it,” Tom Corson, the president of RCA Records, said. “The all-you-can-eat access model is starting to make sense to people. And we expect that free is going to roll into subscription and that is going to be a really huge part of our business.”

The question of whether Spotify is good for artists is considerably more vexed. The service has been dogged by accusations that it doesn’t value musicians highly enough. In 2013, Radiohead’s Thom Yorke memorably called Spotify “the last desperate fart of a dying corpse,” a remark that “saddened” Ek. In July, Taylor Swift wrote in a Wall Street Journal editorial, “In my opinion, the value of an album is, and will continue to be, based on the amount of heart and soul an artist has bled into a body of work.” For Swift, streaming is not much different from piracy. “Piracy, file sharing and streaming have shrunk the numbers of paid album sales drastically, and every artist has handled this blow differently,” she wrote.

In early November, when Swift’s new album, “1989,” was released, her label, Big Machine Records, not only declined to make the album available on Spotify but also removed her entire catalogue from the service. Is this a gesture of artistic solidarity, or, as one insider put it, “a stunt to wring the last drop of blood out of what is a dying model”—i.e., album sales? Swift’s impressive first-week sales of “1989,” which were just under 1.3 million albums, making her the year’s top seller, are still well short of the all-time first-week high, 2.4 million, set by ’N Sync, in 2000. And the sixty-nine-per-cent drop-off in “1989” ’s second-week sales suggests that Swift’s seventy-one million Facebook fans didn’t rush out and buy the album when they couldn’t get it on Spotify. They just streamed whatever was available on YouTube, which pays artists even less than Spotify does, or on other sites. Or they set sail for the Pirate Bay, where the album was also No. 1.

On Spotify, music consumption is “frictionless”—a favorite word of Ek’s. In tech terms, we’ve gone from a world of scarcity to one of abundance. Nothing is for sale, because everything is available. The kind of calculations you make on iTunes, such as “I like this song, but not enough to buy it,” don’t matter. It is a music nerd’s dream, which may be why the user population on Spotify tends to lie outside the mainstream. On Spotify, the Pixies’ top songs have about four times as many streams as Neil Diamond’s biggest hits.

The difference between Spotify and Internet radio services, like Pandora, is that Spotify is interactive. You can sample the complete catalogue of most artists’ recordings. (Spotify also has a non-interactive radio component.) Spotify now has some twenty million songs on the service, and twenty thousand new ones are added every day. If you are a “lean forward” listener—that is, the kind of motivated fan who takes the time to discover the music you want—Spotify is a celestial jukebox. But, for Spotify to continue its rapid growth, it must bring in the “lean backers” Pandora caters to. Spotify tries to do this with playlists. It has staff-curated playlists, and users can also make their own—there are more than a billion on the site. The playlist is the album of the streaming world. Spotify is working on getting its service into car stereos, and is negotiating agreements with automobile companies; one such agreement was announced this week. The power of playlists will only grow.