He was best known as the lead singer in Soundgarden and Audioslave, and also performed the theme song to the 2006 James Bond film Casino Royale.
In a statement, Cornell's representative Brian Bumbery said his death was "sudden and unexpected".
It also said the family would be working closely with the medical examiner to determine the cause and asked for privacy.
Cornell died shortly after playing a concert with Soundgarden in Detroit Wednesday evening.
The show was part of a wider tour and the group had a number of live dates scheduled for later this month.
Cornell was born on 20 July, 1964.
As a solo artist, he released four studio albums - most recently 2015's Higher Truth.
His biggest single in the UK was 2006's You Know My Name, which was the theme song to Casino Royale, which starred Daniel Craig.
The track made him the first male American artist to write and perform the theme song for a James Bond movie.
But he was perhaps most famous as the lead singer of Seattle band Soundgarden, which formed in 1984 and went on to release six studio albums.
Their most successful in the UK was 1994's Superunknown, which reached number four in the singles chart.
In 2001, he joined rock supergroup Audioslave, which consisted of Cornell on lead vocals, together with Rage Against The Machine members Tom Morello (guitar), Tim Commerford (bass) and Brad Wilk (drums). The group disbanded in 2007.