It is suspected that he suffered a heart attack.
The band are best known for their 1978 hit Ever Fallen in Love (With Someone You Shouldn’t’ve).
Shelley and former bandmate Howard Devoto, who later started Magazine, formed the Buzzcocks in Bolton in 1976.
They had a profound effect on independent music after releasing the Spiral Scratch EP in 1977 - credited as the first punk record to be released without record company backing.
In a statement released via Twitter, the group's management said: "It's with great sadness that we confirm the death of Pete Shelley, one of the UK's most influential and prolific songwriters and co-founder of the seminal original punk band Buzzcocks.
"Pete's music has inspired generations of musicians over a career that spanned five decades and with his band and as a solo artist, he was held in the highest regard by the music industry and by his fans around the world."
Tributes from other artists and public figures have started to pour in for Shelley.
Singer Tracey Thorn tonight said he had been an “amazing songwriter”.
Leeds North West MP Alex Sobel said: "Very sad to hear that Pete Shelley of the Buzzcocks died today.
"Without him we wouldn’t have had so many great bands and the whole post punk alternative scene would have evolved very differently."