The band was formed in 1997 by original members Gunnar Örn Tynes and Örvar Þóreyjarson Smárason, who were joined a year later by twin sisters Gyða and Kristín Anna Valtýsdóttir. According to Kristín, the band's name was not intended to mean anything. In 2002, after the first American tour, Gyða left the band to return to her studies in Reykjavík. Shortly after Gyða left, Ásthildur Valtýsdóttir joined for singing duties and Serena Tideman replaced Gyða on cello, for a European tour. Then the ensemble evolved to include Ólöf Arnalds and Hildur Guðnadóttir. In early 2006, Kristín also left the band, although it was not officially announced until November 23 of that year.
Despite the departure of some of its members múm has remained together as a collective of musicians. Their fourth album, recorded during 2006, was released on 24th of September, 2007, entitled Go Go Smear the Poison Ivy.
In November 2007, múm toured the East Coast with German musician, Hauschka. They returned in Spring 2008 with the same setlist. Both tours included songs from their latest album, Go, Go Smear the Poison Ivy.
On August 27, 2008, múm announced on their official website that "múm is quietly but surely [working on] their new album. No release date has been etched in stone, but every day will bring it closer." múm also released several pictures of themselves during the recording process on their MySpace page.
On May 22, 2009, during a concert in Burgos, Spain, múm played songs from their upcoming album Sing Along to Songs You Don't Know. It will officially be released on August 24, 2009.