Producer Alan Moulder was a prominent collaborator who helped shape their blend of heavy beats and densely–layered guitar tracks set against Halliday's vocals.

[8] The English-born Toni Halliday met Dave Stewart of Eurythmics after he had read a rock magazine interview with her in which she praised his pre-Eurythmics band, The Tourists.

[9] Garcia had played bass guitar as part of Eurythmics' live band in 1983–84 and on two of their studio albums,[10] while Halliday was signed to Stewart's Anxious Records label as a solo artist.

As Curve, Halliday and Garcia released three acclaimed and increasingly successful EPs (Blindfold, Frozen, and Cherry) throughout 1991 on Anxious Records.

[11] The group toured extensively during this period, with Halliday and Garcia being supported on stage by two additional guitarists (Debbie Smith, later of Echobelly, and Alex Mitchell) and a drummer (Steve Monti, formerly of Ian Dury and the Blockheads).

Highlights of Curve's live career included a performance at the 1992 Glastonbury Festival, and a package tour of the United States and Canada with The Jesus and Mary Chain and Spiritualized.

In 1992, the band released the compilation album Pubic Fruit, containing their first three EPs and an extended mix of the single "Faît Accompli".

That coupled with the stressfulness of the tour in support of the record, may have contributed to Halliday and Garcia's decision to disband the group in 1994.

In the same year, Curve collaborated with Paul Van Dyk by reworking the mostly instrumental song "Words" from the album Seven Ways and also adding Halliday's vocals.

In 2003, Toni Halliday collaborated with the Japanese industrial rock band Acid Android on the song "Faults" from the album with the same name.

In 2010, Curve published some of their most important releases as digital downloads on their Bandcamp page, including a new compilation with 39 songs entitled Rare and Unreleased.

Garcia is also member of the bands The Black Holes (with Jo Neale; 2007–present),[17] The Chronologic (2006–present),[18] Inkraktare (with Mark Wallbridge aka Vasko The Pig; 2009–present),[19] The Secret Meeting (with kaRIN and Statik of Collide; 2007–present),[20] KGC (with Sascha Konietzko and Lucia Cifarelli of KMFDM; 2006–present) and Morpheme (2010–present).

[23] Stephen Thomas Erlewine of Allmusic described Curve's style as a "towering monolith of guitar noise, dance tracks, dark goth, and airy melodies".