Sullivan was born in Lambeth, now part of 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 Major (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 1871, John Hollingshead commissioned Sullivan to work with Gilbert to create the burlesque Thespis for the Gaiety Theatre. The show was successful in the context that it was conceived specifically as a Christmas entertainment and as such ran through to Easter 1872. Plans to revise and revive the piece in 1876 were abandoned when Richard D'Oyly Carte's backers demanded a new show for their money and not a revival. The score was subsequently lost, though one number Little Maid of Arcadee was published by Cramer in 1872 and another was later re-used in The Pirates of Penzance. Recent research would seem to indicate that more than just this one number was reused in the later show.
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 thirteen 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 operetta, 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 of operetta that they created is sometimes known as "Savoy opera" as a result.
In 1883, Sullivan was knighted 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 operettas 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. Gilbert wrote the more serious Yeomen to satisfy Sullivan's urge for grand opera, and while Sullivan was pacified for a time, in 1890 he 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 .