Memphis, Tennessee is a Chuck Berry song, one of his most popular and commercially successful numbers. It came out in 1959 as a single, the flip side of Back in the USA, which was a number one hit.
In 1963 Chuck Berry's version went to number six in the UK. The Johnny Rivers cover of the song in 1964 (based on a slightly different instrumental arrangement) went to number 2 in the US. In addition to the Beatles' famous cover of this song, it was recorded by The Animals, The Grateful Dead, Count Basie, Paul Anka, Tom Jones, Jerry Lee Lewis, Elvis Presley, the Rolling Stones, the Ventures, Del Shannon, Conway Twitty, and many other groups and artists.
Memphis, Tennessee is a story song, much like a ballad from centuries past, in which important details are not revealed until close to the end. The singer is urgently trying to get in touch with "Marie." It takes four verses to learn that she is the singer's daughter, and is six years old, and that the girl's mother was, for some reason, trying to keep her incommunicado from her father.
The Beatles performed Memphis, Tennessee regularly during 1961 and 1962. In 1963, the Beatles made five separate recordings of the song for the BBC. The take made on July 10, 1963, for the seventh installment of Pop! Go the Beatles (aired on July 30, 1963), is reproduced in the collection, Live at the BBC.
They recorded it for the January 1, 1962 Decca audition (with Pete Best on the drums) which remains unreleased though available via Bootlegs.