The memorial to the Joy Division frontman was taken from a Macclesfield's cemetery reports the BBC.
Curtis was 23 when he hanged himself in the kitchen of his Macclesfield home after having viewed Werner Herzog's film Stroszek and listening to Iggy Pop's The Idiot in May 1980, shortly before the band were due to go on tour in the US.
He met Bernard Sumner and Peter Hook in 1976 who he would later go on to start Joy Division with. The name "Joy Division" stemmed from the sexual slavery wing of a Nazi concentration camp in the 1955 novel The House of Dolls. Ian's poetry and lyrics were filled with imagery of emotional isolation, death, alienation, and urban decay.
The headstone is visited by thousands of fans every year, paying their respects to the singer who committed suicide in 1980 at the age of 23.
Officers are appealing for anyone with information on its whereabouts to contact them.
Detectives said the stone, which has the inscription "Ian Curtis 18 - 5 - 80" and the words "Love Will Tear Us Apart" was taken sometime between Tuesday afternoon and Wednesday morning.
A police spokesman said: "There is no CCTV in the area and there are no apparent leads as to who is responsible for the theft.
"This is a very unusual theft and I am confident that someone locally will have knowledge about who is responsible or where the memorial stone is at present.
My Trusted MOGs
What drunken nutbag lugged off with an entire tombstone?
My Trusted MOGs
Does not anyone respect anything? Geeze!
My Trusted MOGs
I guess some people just want a piece of him that badly, its sad really
My Trusted MOGs
Let me at 'em...let me at 'em! Grrrr
My Trusted MOGs
I started laughing when I first read your title Adam! How ridiculous is this? Some people just have no respect OR boundaries!!!!
My Trusted MOGs
Can't imagine it was light.
My Trusted MOGs
We can rule Carmen Sandiego out. This monument was too small for her to steal.
My Trusted MOGs
Too bad -- the bit about him listening to Iggy Pop's The Idiot prior to hanging himself was news to me. I used to listen to that bad boy on 8-Track! Great version of Bowie's "China Girl" on there, before Bowie released it himself!
My Trusted MOGs
For fuck's sake.
It wasn't me, btw.