Donald Swann was born in Llanelli, Wales. His father was a Russian doctor of English descent, from the expatriate community that started out as the Muscovy Company, and his mother was a nurse from Transcaspia ... they were refugees from the Russian Revolution. Swann's great-grandfather, Alfred Trout Swan, a draper from Lincolnshire, emigrated to Russia in 1840 and married the daughter of the horologer to the Tsars. Sometime later the family acquired a second 'n' to their surname. The family moved to London, where Swann attended Dulwich College Preparatory School and Westminster School (where he first met Michael Flanders).

In 1941 Swann was awarded an exhibition to Christ Church, Oxford to read modern languages. In 1942 he registered as a conscientious objector and served with the Friends' Ambulance Unit (a Quaker relief organisation) in Egypt, Palestine and Greece. After the war, Swann returned to Oxford to read Russian and Modern Greek.

Donald Swann was married twice; he married Janet Oxborrow in 1955 and they were divorced in 1983; his second wife was the art historian Alison Smith. In 1992 he was diagnosed with cancer. He died at Trinity Hospice in South London on 23 March 1994, survived by both wives and two children from his first marriage: Rachel and Natasha.

A chance meeting between Swann and Flanders in 1948 led to the start of their professional partnership. They began writing songs and light opera, Swann writing the music and Flanders writing the words. Their songs were performed by artists such as Ian Wallace and Joyce Grenfell. They subsequently wrote two two-man revues, At the Drop of a Hat and At the Drop of Another Hat, which they performed all over the world until their partnership ended in 1967.

At the same time, Swann was maintaining a prolific musical output, writing the music for several operas and operettas, including a full-length version of C. S. Lewis's Perelandra, and a setting of J. R. R. Tolkien's poems from The Lord of the Rings to music in The Road Goes Ever On collection. A life-long friendship with Sydney Carter resulted in scores of songs, the best known being "The Youth of the Heart" which reappeared in At the Drop of A Hat, and a musical Lucy & the Hunter. After his partnership with Flanders ended, Swann continued to give solo concerts and to write for other singers. He also formed the Swann Singers and toured with them in the 1970s. Throughout the 1980s and early 90s he continued performing in various combinations with singers and colleagues and as a solo artist. In the later years of his life he 'discovered' Victorian poetry and composed some of his most profound and moving music to the words of William Blake, Emily Dickinson, Christina Rossetti, Oscar Wilde and others. He wrote a number of hymn tunes which appear in modern standard hymn books.

It is estimated that Swann wrote or set to music nearly 2,000 songs during his career.