In a statement, Carver County Sheriff Jim Olson said his deputies responded to a medical call at about 09:43 local time (14:43 GMT) and later found an unresponsive adult male in an elevator at Paisley Park Studios.
First responders tried to revive him with CPR but he was pronounced dead at 10:07.
Hundreds of fans gathered outside Paisley Park. US President Barack Obama said the world had lost a "creative icon".
'A visionary' - Madonna
Born in 1958, Prince was a prolific writer and performer from a young age - reportedly writing his first song when he was seven.
A singer, songwriter, arranger and multi-instrumentalist, Prince recorded more than 30 albums. His best known hits include Let's Go Crazy and When Doves Cry.
'A joy to watch': By Mark Savage, BBC entertainment reporter
A musical prodigy from a broken home, Prince famously wrote, arranged, produced and played almost all of his hit records.
But the Purple man's purple patch really came with his first band The Revolution.
With them by his side, he wrote more than two dozen rock classics in a five-year flurry.
Purple Rain, Little Red Corvette, 1999, Raspberry Beret, When Doves Cry, Kiss... At the same time, he dashed off Manic Monday for The Bangles and Nothing Compares 2U, made famous by Sinead O'Connor.
In the studio, he was unstoppable. But the magic really happened on stage. He would vamp, preen and tease an audience into a frenzy, then slay them with a quiet moment of crystalline beauty. He was a joy to watch.
He also wrote music for several artists - Sinead O'Connor's version of Nothing Compares 2U became a worldwide smash in 1990.
In 1984, he won an Oscar for the score to Purple Rain, a film in which he also starred.
Throughout his career he had a reputation for secrecy and eccentricity, once changing his name to an unpronounceable symbol.
In 2004, Prince was inducted into the Rock and Roll Hall of Fame, which said he "rewrote the rulebook".
He had a mercurial relationship with technology. In 2000, he released singles via the pioneering music-sharing service Napster, but he later declared the internet "completely over" and refused to allow his music on major streaming platforms.
Prince's latest album, HITnRUN Phase Two, was released last year and he had been touring as recently as last week.
On 15 April he was taken to hospital after his private plane made an emergency landing in Illinois. It happened just hours after he had performed on stage in Georgia. He was treated and released after a few hours.
Tributes have been pouring in from artists young and old, across the musical spectrum.
Madonna, who dated Prince briefly, described him as a "visionary who changed the world"
Chic guitarist Nile Rogers said there were "tears and love on our tour bus"
Singer Justin Timberlake: "Numb. Stunned. This can't be real"
Guitarist Slash said Prince was "one of the greatest musical talents of my lifetime. Maybe of the 20th century"
Boy George: "Today is the worst day ever. Prince RIP I am crying!"
"I can't believe it, I'm in total shock. So many wonderful memories," wroteLionel Richie
Musician and actor Wyclef Jean: "RIP to the King Prince thank u for inspiring me to be a Musician First and using this tool to heal people"
Mick Jagger said Prince's talent was "limitless", calling him a "revolutionary artist, a great musician, a wonderful lyricist"
"It's such a blow. It's really surreal. It's just kind of unbelievable," Aretha Franklin told MSNBC. "He was definitely an original and a one of a kind. Truly there was only one Prince."