Fife Police confirmed to the BBC: "The death is non-suspicious and due to a medical matter."
Wilson, a heavy smoker, was diagnosed with the lung disorder chronic pulmonary obstructive disease in 2009.
He won the 1982 and 1989 World Championships and reached a minimum of the quarter-finals at every tournament between 1979 and 1991.
He also won the British Championship on four occasions.
The Professional Darts Corporation tweeted: "We are very saddened to hear of the death of former World Champion Jocky Wilson. Simply devastated. #RIPJocky"
Thirteen-times champion Phil Taylor told the Sun on Sunday: "We owe Jocky so much. Jocky really was a true legend."
Bobby George, who twice finished as runner-up in the World Championship, also paid tribute.
"Farewell John Thomas (Jocky) Wilson," he tweeted. "We had some great times together will never forget you and your toothless grin! Great #darts #legend"
Wilson turned professional in 1979 and retired in 1996.