Groves studied music as a chorister at St. Paul's Cathedral Choir School and later enrolled at the Royal College of Music, where he focused on piano and organ studies and choral and orchestral music, leading many ensembles in both venues during his student years. He worked with Toscanini in performances of Beethoven's Missa Solemnis and the Brahms and Verdi requiems. In 1938, the BBC Theatre Chorus engaged Groves to serve as a chorus master. Six years later, he was appointed conductor of the Manchester-based BBC Northern Orchestra (later named the BBC Philharmonic), his first such permanent assignment and then left the BBC NO in 1951 and assumed duties as conductor of the Bournemouth Municipal Orchestra the following year. The group's repertory was extended to include opera, performed in conjunction with the Welsh National Opera, and the orchestra's name was changed in 1954 to the Bournemouth Symphony Orchestra. His fine work with the Welsh National Opera and with operatic repertory in general did not go unnoticed: in 1961, he was appointed director of the Welsh National Opera, becoming the first conductor to hold the post in a full-time capacity. Once more, Groves expanded the repertory of his new ensemble, adding Wagner's Lohengrin and other major works previously ignored. Not only did Groves divulge an adventurous streak in his inclusion of new works, but he quite consistently brought them off with much acclaim. In 1963, Groves became the permanent music director of what would become the most-respected British ensemble not based in London, the Royal Liverpool Philharmonic. He took the group on a highly acclaimed tour of Germany and Switzerland in 1966. The following year, Groves took on a concurrent assignment as associate conductor of the London-based Royal Philharmonic Orchestra. Further tours for both his groups ensued in the next few years. With the Royal Liverpool Philharmonic Orchestra, he returned to Germany and Switzerland in 1968 to critical acclaim, and in 1970 toured Poland, where he scored a notable triumph at the Festival of Modern Music (Warsaw); with the RPO, he toured the United States to generally enthusiastic receptions. In 1973, Groves was knighted; five years later, he was appointed music director of the English National Opera, without doubt attaining one of the most coveted positions in British opera. For the last decade or so of his career, Groves guest-conducted ensembles throughout the world.
Of his numerous recordings, many are still available on a variety of labels. Not surprisingly, he displayed a fondness in both concert and recording venues for the music of Delius, Vaughan Williams, Arnold, Maxwell Davies, and of other British composers., Rovi