The son of an actor father, and a mother who had been a concert-violinist before she married, Michael Flanders (b. 1 March 1922, London, England, d. 14 April 1975), was brought up in a musical household. He learned to play the clarinet and made his stage debut at the age of seven in a singing contest with Uncle Mac's Minstrel Show. At Westminster School in London, where Peter Ustinov was one of his classmates, he started to write and stage revues. His search for a pianist led him to Donald Swann (b. Donald Ibrahim Swann, 30 September 1923, Llanelli, Wales, d. 23 March 1994, London, England), and their first revue together was Go To It. At Oxford University in 1940 Flanders played in and directed several productions for the Dramatic Society and made his professional debut as Valentine in Shaw's You Never Can Tell, at the Oxford Playhouse. In 1943, while serving in the Royal Navy Volunteer Reserve, having survived the infamous convoys to Russia, he was struck down by poliomyelitis. Three years later he was discharged from hospital, in a wheelchair, and with a full beard which he retained for the rest of his life.

Unable to resume a normal acting career, Flanders turned to writing and broadcasting. He contributed lyrics to several West End revues, in collaboration with Swann, including Penny Plain (1951), Airs On A Shoestring (1953) and Fresh Airs (1956). Flanders also appeared extensively on radio, and later, television, in programmes ranging from sports commentary to poetry readings, and including a spell of two years as chairman of The Brains Trust. His translation of Stravinsky's Soldier's Tale (with Kitty Black) became the standard English version, and his concert performance of it with Peter Ustinov and Sir Ralph Richardson was a surprise sell-out at the Royal Festival Hall in 1956. After successfully entertaining their friends at parties with their own songs, Flanders and Swann decided to perform professionally, so on New Year's Eve 1956, they opened their own two-man show, At The Drop Of A Hat, at the intimate New Lindsey Theatre, Notting Hill, west London, moving three weeks later into the West End's Fortune Theatre. The show was a smash hit and ran for over two years. It was reported that Princess Margaret attended a performance, and returned the following week with the Queen and the Duke of Edinburgh. With Flanders" urbane image contrasting with Swann's almost schoolboy enthusiasm, they introduced songs such as "The Hippopotamus ("Mud, Mud, Glorious Mud')", "Misalliance", "A Gnu", and "Madeira M'Dear?". Two albums from the show were released, the earlier mono recording being preferable to the later stereo issue from the last night of the London run. In 1959 the show opened on Broadway, billed as "An After-Dinner Farrago", and later toured the USA, Canada and the UK. In 1963 at the Haymarket Theatre, London, they presented a fully revised version entitled At The Drop Of Another Hat, which included songs such as "The Gas-Man Cometh", "First And Second Law" and "Bedstead Men". During 1964/5 they toured Australia, New Zealand and Hong Kong, before returning to the West End in 1965, and yet again, to New York in the following year.

Meanwhile, Flanders was still continuing with his other work, writing, broadcasting and performing theatrical speech recitals. He published Creatures Great And Small, a children's book of verses about animals and, together with Swann, released an album of animal songs entitled The Bestiary Of Flanders And Swann. Flanders was awarded the OBE in 1964. After the partnership broke up in 1967, Swann, who was born of Russian parents, continued to compose. In the 50s he had written the music for revues such as Pay The Piper and Wild Thyme, but in later years his music reflected his religious beliefs (he was a Quaker) and his love of Greece, and many other interests. He was still working right up to the time he died from cancer in 1994. In that same year, a musical celebration of the works of Michael Flanders and Donald Swann, entitled Under Their Hats, was presented at the King's Head Theatre in London.