Sullivan was born in London. His father was a military bandmaster, and by the time Arthur had reached the age of 8, he was proficient with all the instruments in the band. Following a stay at private school in Bayswater, he was admitted to the choir of the Chapel Royal, attending its school in Cheyne Walk. While there, he began to compose anthems and songs. In 1856, he received the first Mendelssohn prize and became a student at the Royal Academy of Music until 1858.
In 1858, Sullivan travelled to Leipzig, where he continued his studies and took up conducting. He credited this period with tremendous musical growth, and his return to London in 1862 saw the production of his incidental music to Shakespeare's The Tempest performed at the Crystal Palace. He began building a reputation as Britain's premier composer, and 1866 saw the first performance of his Symphony in E (Irish). Other pieces from this period include the overture In Memoriam (1866), the oratorio The Prodigal Son (1869), the well-known tune to the hymn Onward Christian Soldiers (1872) and the song The Lost Chord (1877).
In 1866, he supplemented his income by producing the musical score to a one act operetta, Cox and Box. This led to his most famous and lucrative works as a composer for the musical theatre.
In the autumn of 1867, he travelled with Sir George Grove to Vienna, returning with a treasure-trove of undiscovered Schubert scores.
In 1871, John Hollingshead commissioned Sullivan to work with Gilbert to create the operetta Thespis for the Gaiety Theatre. The show was not particularly successful and the score was subsequently lost, though one number was later re-used in The Pirates of Penzance.
Gilbert and Sullivan's real collaborative efforts began in 1875 when Richard D'Oyly Carte commissioned them to write a one act piece, Trial by Jury. Its success was so great that the three men formed an often turbulent partnership which lasted for twenty years and fourteen operettas. Trial was followed in 1877 by The Sorcerer, and in 1878 by their greatest success so far, HMS Pinafore. This last was much pirated in America, and in 1879, Gilbert and Sullivan crossed the Atlantic to protect their copyrights, producing The Pirates of Penzance in New York.
The next Gilbert and Sullivan opera, Patience, opened in the Opéra Comique, London in 1881 and was transferred to the specially-built Savoy Theatre later the same year. All the duo's subsequent collaborations, which include Iolanthe (1882), The Mikado (1885) and The Yeomen of the Guard (1888), opened there, and the genre they created became known as "Savoy Opera" as a result.
In 1883, Sullivan was knighteded by Queen Victoria. Contemporary critics felt that this should put an end to his career as an operetta composer, believing that a musical knight should not stoop below the level of oratorio or "grand opera". Sullivan too, despite the financial security the Savoy operas gave him, increasingly viewed his work with Gilbert as unimportant and beneath his skills. Furthermore he was unhappy that he was having to tone down his music to ensure that Gilbert's words could be heard. In 1886, Sullivan went some way to appeasing his critics by the production of the cantata The Golden Legend, which he and most of his contemporaries considered his masterpiece. Finally, in 1890, Sullivan broke away acrimoniously from Gilbert following the production of The Gondoliers and, with D'Oyly Carte, produced his only grand opera, Ivanhoe, at the new English Opera House. Subsequently however he returned to work with Gilbert on two more operettas and wrote three more with other collaborators.
Sullivan, who had suffered from ill health throughout his life, succumbed to pneumonia at his house in London on November 22, 1900. A monument in his memory was erected in the Victoria Embankment Gardens.