There are at least two groups using this name:
1) All That Jazz was originally a temporary group commissioned by Studio Ghibli to create jazz-style covers of soundtracks from Japan’s Studio Ghibli animated films. They went on to make covers of popular Japanese music and anime theme songs. Much of All That Jazz’s music is purely instrumental, but COSMIC HOME's Yuriko Kuwahara (桑原由里子, クワハラユリコ) and Naomi Hattori (服部直美) provide vocals on many tracks. Only Kuwahara sings on the Studio Ghibli covers, while their other cover albums feature both Hattori and Kuwahara.
2)A Swedish new wave/post punk band active from 1981 - 1991.
Members: Peter Lööf (vocals & guitar), Patrik Willard (bass), Per Birger Ejnelind (guitar), Pontus Wärmhed (drums), Niklas Hellberg (keyboards), Marie Lindbäck (vocals)
Major Influences: Joy Division, The Cure, Bauhaus
All That Jazz was formed in Karlstad, Sweden in 1981 by Peter, Patrik, and Per, who had all been members of punk bands previously. Their first single Banner of Love was released in January 1983 on Wire Records. The band was signed by major label Virgin Records and released a self-titled album produced by top British new wave producers Clive Langer and Alan Winstanley in 1987. Before their second album, Color Blind, was released in 1990, band members Patrik and Marie both left the band. The band continued to suffer from internal strife, the tours for Color Blind were cancelled, and All That Jazz disbanded. .