Among Radiohead's earliest influences were Queen, Pink Floyd and Elvis Costello, post-punk acts such as Joy Division, Siouxsie and the Banshees and Magazine, and significantly 1980s alternative rock bands such as U2, R.E.M., the Pixies, the Smiths and Sonic Youth.

The members of Radiohead met while attending Abingdon School, an independent school for boys in Abingdon, Oxfordshire. Guitarist and singer Thom Yorke and Bassist Colin Greenwood were in the same year, Guitarist Ed O'Brien and Drummer Philip Selway the year above, and multi-instrumentalist Jonny Greenwood, brother of Colin, two years below. In 1985, they formed On a Friday, the name referring to the band's usual rehearsal day in the school's music room. Jonny was the last to join, first on harmonica and then keyboards, but soon became the lead guitarist; he had previously been in another band, Illiterate Hands, with musician Nigel Powell and Yorke's brother Andy Yorke. According to Colin, the band members picked their instruments because they wanted to play music together, rather than through an interest in the particular instrument: "It was more of a collective angle, and if you could contribute by having someone else play your instrument, then that was really cool." At one point, On a Friday featured a saxophone section.

Although all but Jonny had left Abingdon by 1987 to attend university, On a Friday continued to rehearse on weekends and holidays. At the University of Exeter, Yorke played with the band Headless Chickens, performing songs including Future Radiohead material, and met Artist Stanley Donwood, who would later create artwork for the band. In 1991, On a Friday regrouped, sharing a house on the corner of Magdalen Road and Ridgefield Road, Oxford.

Yorke is Radiohead's principal Songwriter and lyricist; songs usually begin with a Sketch by Yorke, which is harmonically developed by Jonny Greenwood before the others develop their parts. Arrangement is a collaborative effort, with all the band members having roles in the process; all songs are credited to "Radiohead". The band often try several approaches to songs; for Example, "Nude" was first performed in the 1990s, but was not released until 2007 in a radically different arrangement. Similarly, the band took over two decades to settle on the arrangement for "True Love Waits", released in 2016. In 2017, Jonny Greenwood said he saw Radiohead as "just a kind of an arrangement to form songs using whatever Technology suits the song. And that Technology can be a cello or it can be a laptop. It's all sort of machinery when looked at in the right way."

As On a Friday continued to perform in Oxford, including more performances at the Jericho Tavern, record labels and producers became interested. Chris Hufford, Slowdive's Producer and co-owner of Oxford's Courtyard Studios, attended an early On a Friday concert at the Jericho Tavern. Impressed, he and his partner Bryce Edge produced a demo tape and became On a Friday's managers; they remain Radiohead's managers today. In late 1991, after a chance meeting between Colin and EMI A&R representative Keith Wozencroft at Our Price, the record shop where Colin worked, On a Friday band signed a six-album recording contract with EMI. At the label's request, the band changed their name; "Radiohead" was taken from the song "Radio Head" on the Talking Heads album True Stories (1986).

Radiohead recorded their debut release, the Drill EP, with Chris Hufford and Bryce Edge at Courtyard Studios. Released in May 1992, its chart performance was poor. The band enlisted Paul Kolderie and Sean Slade, who had worked with US indie bands Pixies and Dinosaur Jr., to produce their debut album, recorded quickly in an Oxford studio in 1992. With the release of the "Creep" single later that year, Radiohead began to receive attention in the British music press, not all of it favourable; NME described them as "a lily-livered excuse for a rock band", and "Creep" was blacklisted by BBC Radio 1 because it was deemed "too depressing".

Unexpected attention for the single in the US prompted EMI to improvise new promotional plans, and the band shuttled back and forth between continents, playing more than 150 concerts in 1993. Radiohead nearly broke up due to the pressure of sudden success as the Pablo Honey supporting tour extended into its second year. The band members described the tour as difficult to adjust to, saying that towards its end they were "still playing the same songs that [they had] recorded two years previously ... like being held in a time warp", when they were eager to work on new songs.

Graphic Artist Stanley Donwood met Yorke when both were art students, and with Yorke has produced all of Radiohead's album covers and visual artwork since 1994. Donwood works in the studio with the band as they record, allowing the music to influence the artwork. He and Yorke won a Grammy in 2002 for the special edition of Amnesiac packaged as a library book.

By late 1995, Radiohead had already recorded one song that would appear on their next record. "Lucky", released as a single to promote the War Child charity's The Help Album, was recorded in a brief session with Nigel Godrich, the young audio Engineer who had assisted on The Bends and produced a 1996 B-side, "Talk Show Host". The band decided to self-produce their next album with Godrich, and began work in early 1996. By July they had recorded four songs at their rehearsal studio, Canned Applause, a converted apple shed in the countryside near Didcot, Oxfordshire.

In August 1996, Radiohead toured as the opening act for Alanis Morissette. They resumed recording not at a studio but at St. Catherine's Court, a 15th-century mansion near Bath. The sessions were relaxed, with the band playing at all hours of the day, recording in different rooms, and listening to the Beatles, DJ Shadow, Ennio Morricone and Miles Davis for inspiration.

Radiohead were largely inactive following their 1997–1998 tour; after its end, their only public performance in 1998 was at an Amnesty International concert in Paris. During the period the band came close to splitting up, and Yorke developed severe depression. In early 1999, Radiohead began work on their next album. Although the success of OK Computer meant there was no longer any pressure or a deadline from their record label, tension was high. Band members had different visions for Radiohead's Future, and Yorke experienced writer's block, influencing him toward more abstract, fragmented songwriting. Radiohead secluded themselves with Producer Nigel Godrich in studios in Paris, Copenhagen, and Gloucester, and in their newly completed studio in Oxford. Eventually, the members agreed on a new musical direction, redefining their instrumental roles. After nearly 18 months, Radiohead's recording sessions were completed in April 2000.

In October 2000 Radiohead released their fourth album, Kid A, the first of two albums from these recording sessions. A departure from OK Computer, Kid A featured a minimalist and textured style with more diverse instrumentation, including the ondes Martenot, programmed electronic beats, strings, and jazz horns. It debuted at number one in many countries, including the US, where it became the first Radiohead album to debut atop the Billboard chart and the first US number-one album by any UK act since the Spice Girls in 1996. This success was attributed variously to marketing, to the album's leak on the file-sharing network Napster a few months before its release, and to advance anticipation based, in part, on the success of OK Computer. Although Radiohead released no singles from Kid A, promos of "Optimistic" and "Idioteque" received radio play, and a series of "blips", or short videos set to portions of tracks, were played on music channels and released free online. Inspired by Naomi Klein's anti-globalisation book No Logo, Radiohead continued a 2000 tour of Europe in a custom-built tent free of advertising; they also promoted Kid A with three sold-out North American theatre concerts.

Radiohead's fifth album, Amnesiac, was released in June 2001. It comprised additional tracks from the Kid A recording sessions, plus one track recorded after Kid A's release, "Life in a Glasshouse", featuring the Humphrey Lyttelton Band. Radiohead stressed that they saw Amnesiac not as a collection of B-sides or "leftovers" from Kid A but an album in its own right. It topped the UK Albums Chart and reached number two in the US, and was nominated for a Grammy Award and the Mercury Music Prize. Radiohead embarked on a world tour, visiting North America, Europe and Japan. "Pyramid Song" and "Knives Out", Radiohead's first singles since 1998, were modestly successful. A live album, I Might Be Wrong: Live Recordings, released in November 2001, features performances of seven songs from Kid A and Amnesiac, and the previously unreleased acoustic track "True Love Waits".

In July and August 2002, Radiohead toured Portugal and Spain, playing a number of new songs. For their next album, the band sought to explore the tension between human and machine-generated music and capture a more immediate, live sound. They and Godrich recorded most of the material in two weeks at Ocean Way Recording in Los Angeles, with the rest of the album recorded in Oxford into the next year. The band described the recording process as relaxed, in contrast to the tense sessions for Kid A and Amnesiac.

Radiohead's sixth album, Hail to the Thief, was released in June 2003, combining guitar rock with electronic music. Its lyrics were influenced by what Yorke called "the general sense of ignorance and intolerance and panic and stupidity" following the 2000 election of US President George W. Bush. The album was promoted with a website, Radiohead.tv, where short films, music videos and live webcasts from the studio were streamed at scheduled times. Hail to the Thief debuted at number one in the UK and number three on the Billboard chart, and was eventually certified platinum in the UK and gold in the US. The singles "There There", "Go to Sleep" and "2 + 2 = 5" achieved heavy circulation on modern rock radio. At the 2003 Grammy Awards, Radiohead were again nominated for Best Alternative Album, and Producer Godrich and Engineer Darrell Thorp received the Grammy Award for Best Engineered Album. In May 2003, Radiohead embarked on a world tour and headlined Glastonbury Festival for the second time. The tour finished in May 2004 with a performance at the Coachella Festival in California. A compilation of Hail to the Thief B-sides, remixes and live performances, Com Lag (2plus2isfive), was released in April 2004.

Following the Hail to the Thief tour, Radiohead went on hiatus to spend time with their families and work on side projects. Yorke and Jonny Greenwood contributed to the Band Aid 20 single "Do They Know It's Christmas?", produced by Godrich. Greenwood composed soundtracks for the films Bodysong (2004) and There Will Be Blood (2007); the latter was the first of several collaborations between Greenwood and Director Paul Thomas Anderson. In July 2006, Yorke released his debut solo album, The Eraser, comprising mainly electronic music. He told Pitchfork: "I've been in the band since we left school and never dared do anything on my own ... It was like, 'Man, I've got to find out what it feels like,' you know?"

Radiohead began work on their seventh album in February 2005 with no record label. In an effort to "get out of the comfort zone", they decided against involving Godrich, with whom they had recorded five albums, and hired Producer Spike Stent. The collaboration with Stent was unsuccessful and ended in April 2006. In September 2005, Radiohead contributed "I Want None of This", a piano dirge, for the War Child charity album Help: A Day in the Life. The album was sold online, with "I Want None of This" the most downloaded track, though it was not released as a single. In late 2006, after touring Europe and North America with new material, the band re-enlisted Godrich and resumed work in London, Oxford and rural Somerset, England. Recording ended in June 2007 and the recordings were mastered the following month.

In Rainbows was released physically in the UK in late December 2007 on XL Recordings and in North America in January 2008 on TBD Records, charting at number one both in the UK and in the US. The record's Retail success in the US – after having been legally available for months as a free download – was Radiohead's highest chart success in that country since Kid A. It became their fifth UK number-one album and sold more than three million copies in one year. The album received critical acclaim for its more accessible sound and personal lyrics. It was nominated for the short list of the Mercury Music Prize and went on to win the 2009 Grammy Award for Best Alternative Music Album. Their production team won the Grammy for Best Boxed or Special Limited Edition Package, while Radiohead received their third nomination for Album of the Year. Along with three other nominations for the band, Godrich's production and the "House of Cards" music video also received nominations. Yorke and Jonny Greenwood performed "15 Step" with the University of Southern California Marching Band at the televised award show.

In June 2008, EMI released a greatest hits album, Radiohead: The Best Of. It was made without Radiohead's input and only contains songs released under their recording contract with EMI. Yorke was critical of the release, saying: "There's nothing we can do about it. The work is really public property now anyway, in my head at least. It's a wasted opportunity in that if we'd been behind it, and we wanted to do it, then it might have been good." In August 2008, EMI reissued "special editions" of Radiohead's back catalogue as part of its "From the Capitol Vaults" series. From mid-2008 to early 2009, Radiohead toured North America, Europe, Japan and South America to promote In Rainbows, and headlined the Reading and Leeds Festivals in August 2009.

In May 2009, Radiohead began new recording sessions with Godrich. In August, they released "Harry Patch (In Memory Of)", a tribute song to Harry Patch, the last surviving British soldier to have fought in World War I, with proceeds donated to the British Legion. The song has no conventional rock instrumentation, and instead comprises Yorke's vocals and a string arrangement composed by Jonny Greenwood. Later that month, another new song, "These Are My Twisted Words", featuring krautrock-like drumming and guitars, was leaked via torrent, possibly by Radiohead themselves. It was released as a free download on the Radiohead website the following week. Commentators saw the releases as part of Radiohead's new unpredictable release strategy, without the need for traditional marketing campaigns.

That year, Yorke formed a new band to perform The Eraser live, Atoms for Peace, with Musicians including Godrich and Red Hot Chili Peppers Bassist Flea; the band played eight North American shows in 2010. In January 2010, Radiohead played their only full concert of the year in the Los Angeles Henry Fonda Theater as a benefit for Oxfam. Tickets were auctioned, raising over half a million US dollars for the NGO's 2010 Haiti earthquake relief. In June, Yorke and Jonny Greenwood performed a surprise set at Glastonbury Festival, performing Eraser and Radiohead songs. On 30 August, Selway released his debut solo album, Familial. In September 2010, Radiohead released the soundboard recording of their 2009 Prague performance for use in a fan-made concert video, Live in Praha. In December, a fan-made video of Radiohead's Oxfam benefit performance, Radiohead for Haiti, was released via YouTube and torrent with Radiohead's support and a "pay-what-you-want" link to donate to Oxfam. The videos were described as examples of the band's openness to fans and positivity toward non-commercial internet distribution.

Dilly Gent has been responsible for commissioning all Radiohead music videos since OK Computer, working with the band to find a Director for each project. Since Radiohead's formation, Andi Watson has been their lighting and stage Director, designing the visuals of live concerts, such as the carbon-neutral "LED forest" of the In Rainbows tour. Technician Peter "Plank" Clements has worked with Radiohead since before The Bends, overseeing the setup of their instruments for studio recordings and live performances. Drummer Clive Deamer has performed and recorded with Radiohead since 2011.

In September 2012, EMI was bought by Universal Music. The European Commission approved the deal on the condition that Universal Music divest EMI's Parlophone label, which controlled the Radiohead albums recorded under their contract with EMI. In February 2013, Parlophone, along with Radiohead's back catalogue, was bought by Warner Music Group (WMG). As a condition of the purchase, WMG made an agreement with the Merlin Network and trade group Impala to divest 30% of the Parlophone catalogues to independent labels, with Artist approval. In October 2015, Radiohead sued Parlophone for deductions made from downloads of their back catalogue. In April 2016, as a result of the Impala agreement, WMG transferred Radiohead's back catalogue to XL Recordings, who had released the physical editions of In Rainbows and The King of Limbs and most of Yorke's solo work. Radiohead: The Best Of and the "special editions" of Radiohead albums, issued by EMI in 2008 without Radiohead's approval, were removed from streaming services. In May 2016, XL reissued Radiohead's back catalogue on vinyl.

The Kid A and Amnesiac sessions brought a change in Radiohead's music and working methods. Since their shift from conventional rock music instrumentation toward an emphasis on electronic sound, the members have gained flexibility and now regularly switch instruments depending on the particular song requirements. On Kid A and Amnesiac, Yorke played keyboard and bass, while Jonny Greenwood often played ondes Martenot, Bassist Colin Greenwood worked on sampling, and O'Brien and Selway branched out to drum machines and digital manipulation, also finding ways to incorporate their primary instruments – guitar and percussion, respectively – into the new sound. The relaxed 2003 sessions for Hail to the Thief led to a different dynamic, with Yorke admitting that his power in the band had been "absolutely unbalanced" and that he would "subvert everybody else's power at all costs. But ... it's actually a lot more healthy now, democracy-wise."

Radiohead began work on their ninth studio album in September 2014, joined again by Godrich. In 2015 they resumed work in the La Fabrique studio near Saint-Rémy-de-Provence, France. The sessions were marred by the death of Godrich's father, and Yorke's separation from his wife, Rachel Owen, who died of cancer several months after the album's completion. Recording was interrupted when the band were commissioned to write the theme for the 2015 James Bond film Spectre. Their first submission, "Man of War", written before OK Computer, was rejected as it had not been written for the film; among other reasons, it would have been ineligible for the Academy Award for Best Original Song. Their second submission, "Spectre", a "brooding" orchestral song, was rejected as "too dark". Radiohead released "Spectre" on Christmas Day 2015 on the audio streaming site SoundCloud.

Radiohead have maintained close relationships with a number of frequent collaborators. Producer Nigel Godrich made his name with Radiohead, working as an audio Engineer on The Bends and as their Producer on every studio album since. He has been dubbed the "sixth member" of the band, in an allusion to George Martin being called the "Fifth Beatle". In 2016, Godrich said of the collaboration: "I can only ever have one band like Radiohead who I've worked with for this many years. That's a very deep and profound relationship. The Beatles could only have ever had one George Martin; they couldn't have switched producers halfway through their career. All that work, trust, and knowledge of each other would have been thrown out of the window and they'd have to start again."

On 20 August 2017, Yorke and Jonny Greenwood performed a benefit concert in Le Marche, Italy, following the August 2016 Central Italy earthquake. Radiohead collaborated with the film Composer Hans Zimmer to record a new version of the King of Limbs track "Bloom" for the nature documentary series Blue Planet II. The new track, "(ocean) Bloom", features new vocals by Yorke recorded alongside the BBC Concert Orchestra. In October, Radiohead were nominated for the Rock and Roll Hall of Fame, the first time they had been eligible since their debut release 25 years prior. In the same month, Selway released his third solo work, the Soundtrack to the film Let Me Go. In 2018, Jonny Greenwood scored his second film by Lynne Ramsay, You Were Never Really Here, and was nominated for an Academy Award for Best Original Score for his fifth collaboration with Anderson, Phantom Thread. Radiohead began a tour of North and South America in 2018.