A Perfect Circle is an American rock supergroup formed in Los Angeles, California, in 1999 by guitarist Billy Howerdel and Tool vocalist Maynard James Keenan. A Perfect Circle released three of their four studio albums in the early 2000s: their debut Mer de Noms in 2000; a follow-up, Thirteenth Step, in 2003; and an album of radically re-worked cover songs, Emotive, in 2004. Shortly after Emotive's release, the band went on hiatus; Keenan returned to Tool and started up solo work under the band name Puscifer, while Howerdel released a solo album, Keep Telling Myself It's Alright, under the moniker Ashes Divide. Band activity was sporadic in the following years; the band reformed in 2010, and played live shows on and off between 2010 and 2013, but fell into inactivity after the release of their greatest hits album, Three Sixty, and a live album box set, A Perfect Circle Live: Featuring Stone and Echo in late 2013. The band reformed in 2017 to record a fourth album, Eat the Elephant, which was released in 2018. After spending the rest of the year touring in support of the album, the band fell into inactivity until 2024 for a brief tour and one-off song "Kindred".

Prone to downtime due to Keenan's other musical commitments, the band has featured a variety of musicians throughout alternating periods of activity and inactivity, and has changed line-ups on each album, leaving Keenan and Howerdel the only constant members. The original incarnation of the band included Paz Lenchantin on bass, Troy Van Leeuwen on guitar, and Tim Alexander on drums. Alexander, however, only performed a handful of live shows and appeared on one song on the group's debut album before being replaced by Josh Freese. Band collaborator and producer Danny Lohner and bassist Jeordie White were also members for a short period in the early 2000s. The band's current lineup features Smashing Pumpkins guitarist James Iha, bassist Matt McJunkins, and drummer Josh Freese, the latter two also being contributors to the related Puscifer and Ashes Divide projects. Despite the varied cast and numerous lineup changes, the primary roles of creating A Perfect Circle's songs has remained consistent with Howerdel as music composer and Keenan writing lyrics and vocal melodies. The band's studio albums have been generally well received critically and commercially, with their first three studio albums selling 4 million copies collectively as of 2005.

A Perfect Circle was originally conceived by Billy Howerdel, a former guitar technician for Nine Inch Nails, the Smashing Pumpkins, Fishbone and Tool. Howerdel met singer Maynard James Keenan in 1992, when Tool was opening for Fishbone, and the two became friends. Three years later, Keenan offered Howerdel, who was looking for lodging, a room in his North Hollywood home. This provided Howerdel the opportunity to play demos of his music for Keenan. Pleased with what he heard, Keenan remarked, "I can hear myself singing [those songs]." Howerdel originally wanted Cocteau Twins lead singer Elizabeth Fraser to be A Perfect Circle's vocalist, but she was unavailable. Howerdel agreed that Keenan would be a good fit, and A Perfect Circle was formed a short time later. The two rounded out the band's initial lineup with bassist and violinist Paz Lenchantin, former Failure guitarist Troy Van Leeuwen, and Primus drummer Tim Alexander. The band played their first show in Los Angeles at the Viper Club Reception in August 1999 followed by a performance at the Coachella Festival that October, by which time Alexander had been replaced by Josh Freese, who worked with Howerdel on Guns N' Roses recording sessions that would later lead to the album Chinese Democracy. While initially in talks to release the album on Volcano Records – Tool's record label – Keenan stated that they instead chose to go with Virgin Records, who Keenan felt better understood that Keenan meant for the band to be an equally important band to Tool, not a minor side project. After the initial shows and securing a record deal, the band entered the studio to begin work on their first album.

I designed the original logo of a symbol, which outwardly looks like a larger and smaller crescent, but actually holds two perfect circles. I designed the logo purposefully to represent Maynard [James Keenan] and me. I thought the logo represented "3 Libras" with its line, "You don't see me at all." I related to the song – feeling like there's a lot to offer, but invisible. Millions suffer with that feeling every day. You can stare at something for so long before you see it in a new light. The insight from something you discover on your own always means more ... the APC logo has two perfect circles in there; they're just hard to see.

The band's debut album, Mer de Noms (French for "Sea of Names"), was released on May 23, 2000. The album was well-received commercially and critically. It debuted at number four on the Billboard 200, selling 188,000 copies in its first week and making it the highest-selling debut album for a rock band. Mer de Noms was later certified platinum by the RIAA, indicating 1 million copies sold, on October 31, 2000. The release earned the band the "Best Debut Album" award from the California Music Awards. In the album review by Rolling Stone, Pat Blashill wrote that Keenan "added an almost operatic angst to Howerdel's songs" and concluded that "A Perfect Circle sound like a desperate dream of what rock used to be. Maybe that's the point." AllMusic's review expressed that "there's little question that the addicting combination of Keenan's aching voice and Howerdel's accomplished songs and production skills made for one of 2000's best splashes in whatever was left of 'modern rock'."

Promotions began shortly after recording for the album was finished. Initially, they served as the opening act for Nine Inch Nails on the 2000 Fragility v2.0 tour, but subsequently embarked on a number of headlining tours all over the world, touring for around eight months straight. In public appearances with the group, Keenan often wore wigs, carrying over a practice begun with Tool of wearing costumes in order to maintain anonymity in his private life. The album produced three singles as well: "Judith", "3 Libras", and "The Hollow", the latter of which featured Alexander's sole studio contribution to the band. All three performed well commercially; peaking at fourth, twelfth, and fourteenth on the Billboard Mainstream Rock chart respectively.

Activity began slowing down for the band by late 2000, with Keenan returning to Tool to finish the recording of what would be their next album, Lateralus. Keenan returned to A Perfect Circle to tour from January to March 2001, until returning to Tool again for the rest of 2001 to release and tour in support of Lateralus. Initial plans were for Howerdel to collaborate long-distance on writing new A Perfect Circle material with Keenan while he toured with Tool, by sending ideas back and forth to one another, though Keenan found it too difficult to balance both, eventually lessening his role while he focused on Tool. Efforts continued through mid-2002, with Howerdel primarily focusing on writing new material while sporadically working with Van Leeuwen, Lenchantin, and Freese. By June 2002, Van Leeuwen estimated that about 80% of the instrumental material was completed while they waited for Keenan's return. However, lineup changes and disagreements on the album's direction would lead to the album not being released for over a year. Both Lenchantin and Van Leeuwen had been working on solo material during the band's downtime, and then moved on to join other bands. Lenchantin would leave to join Billy Corgan's new start-up band Zwan in April 2002, while Van Leeuwen began touring with Queens of the Stone Age shortly after. Ex-Marilyn Manson bassist Jeordie White, formerly known as Twiggy Ramirez, replaced Lenchantin on bass in January 2003. Shortly after, Van Leeuwen permanently left the band, and long-time band collaborator Danny Lohner filled in temporarily as the second guitarist for wrapping up the album.

Keenan rejoined the sessions in early 2003, and the music began taking a different direction with the new lineup and Keenan's influence. Beyond the new personnel, the dynamic had changed: With Mer de Noms, Howerdel had already written and finalized all of the music, with Keenan just contributing the lyrics and vocals. This time around, Keenan was more active in vetoing or altering musical ideas, determined to make something different than just another hard rock album, in fear of it being redundant. Sessions became tense, with Howerdel preferring the heavier compositions written while waiting for Keenan to return to the band, and sometimes taking offense to Keenan's suggested mellower rearrangements. White would play the role of a mediator between the two; being newer to the band helped him provide an outsider's perspective, helping them to find common ground. The band released their second album, Thirteenth Step, on September 16, 2003. The album debuted even higher than Mer de Noms, charting at number 2 in its opening week and selling over 233,000 copies. The album, and its accompanying new sound, fared well critically as well. The AllMusic review praised the band's new sound, describing it as "moodier, tenser, and more atmospheric (if that is possible) recording than its predecessor ... The wide dynamic swathes that were so prominent on the band's debut are all but absent here. The squalling guitars have taken a backseat to carefully crafted melodies where atmospherics are maximized and pulled taut over the listener. While not a radical departure from Mer de Noms, there is a real progression here.... Lyrically, musically, sonically, the Thirteenth Step is proof positive that mainstream rock has plenty of life and vision left in it."