Yazoo's first album, 1982's Upstairs at Eric's was a smash in the UK, where it went to #2 on the charts. (The band's follow-up, 1983's You and Me Both, peaked at #1). However, Yaz had less mainstream success in the U.S., where they were mainly seen as a dance band. In fact, "Don't Go" did not break the top 100 on the Billboard charts in the U.S., although it did hit #1 on the U.S. Dance charts (and #3 in the UK). However, the video was popular on MTV back in the day. I dig the Rocky Horror vibe of the video, and I think it works quite well with the song.
Yazoo disbanded after their 1983 album. Vince Clarke would of course go on to form Erasure (ERV featured "A Little Respect" back in March) while Alison Moyet had a successful solo career (particularly in the UK). Clarke and Moyet re-formed Yazoo in 2008, and they have worked together from time to time since then.
Cool trivia fact: Yazoo never had a top 40 hit in the U.S., although "Only You" and "Situation" both broke the top 100.