Buck has used the guitar since 1982
REM are offering a reward for the return of guitarist Peter Buck's Rickenbacker guitar, which was stolen following a concert in Finland.
In a statement on their website, REM said it was taken from the stage after the gig in Helsinki on Tuesday.
Buck has used the guitar both live and in the studio since the band released their debut EP Chronic Town in 1982.
"Needless to say, this instrument means a great deal to Peter and to REM," the band said.
They added: "It is our sincere hope that whoever is responsible for the loss, or whoever has information which can help recover the guitar, will come forward to receive a generous reward, no questions asked."
REM are currently in the middle of a world tour.
In April, their first studio album in four years, Accelerate, shot to the top of the UK chart.
Last month, rock band Iron Maiden asked their fans to be on the lookout for a guitar stolen after a concert in Athens.
The British group offered a signed tour jacket and a letter of thanks from guitarist Adrian Smith as a reward.