Alexander Gibson was born in Motherwell in Scotland on 11 February 1926. He commenced his conducting career in opera, and was resident conductor at the Vienna State Opera for a time in the 1950s, and subsequently several other continental houses. From 1959 until 1984 he was principal conductor and artistic director of the Scottish National Orchestra. He also held appointments with BBC orchestras. He founded Scottish Opera in 1962, was music director there until 1986, after which he was appointed conductor laureate.
Gibson had an affinity to Scandinavian music, and was awarded the Sibelius medal in 1978. He was president of the Royal Scottish Academy of Music and Drama from 1991 until his death on 14 January 1995. In his memory, the Alexander Gibson School of Opera was founded at the RSAMD in 1998.