Bésame Mucho was written by Mexican songwriter Consuelo Velázquez in 1940, and became a worldwide standard in many languages, including English. It was translated into English by Sunny Skylar.
The Beatles had been singing this song in their early performance repertoire, and manager Brian Epstein chose it as one of the songs to be recorded in January of 1962 for the Decca audition. In June, the Beatles performed this song for George Martin as part of the Parlophone auditions in the Abbey Road Studios. The accounts of that famous day indicate that Martin was not impressed with certain aspects of the Beatles' performance, but he was intrigued overall and later said it was "love at first sight." In fact, Martin saw great raw material, but a need for lots of polishing. Bésame Mucho was just one of the songs that made this clear. For example, you can hear Paul adding frivolous punctuations of "Cha-cha-boom!" in the background of the recording. Paul sings lead, drummer Pete Best is silent, and John and George do backup vocals. Their youth and relative inexperience is audible in this recording.
It merits comparison with a June 6, 1962 recording during the first of their EMI sessions, also featuring Pete Best on drums. This recording was lost and didn't resurface until the 1980s. As a recorded track it did not find its way into the public's hands until Anthology 1 came out in November of 1995. The original recording of Bésame Mucho was mixed in June of 1962 and subsequently aired in a BBC broadcast. The words are sung in English, except for the title, which, mercifully, remains in Spanish.
An impromptu performance on January 29, 1969 during the Let it Be recording sessions and can be seen in the Let It Be film.
An early, low-grade recording from a live performance at Hamburg, Germany's Star-Club in late December of 1962 was released on Live! at the Star-Club in Hamburg, Germany; 1962, a controversial record released on April 8, 1977, making it the first time a Beatles rendition of the song was released for public consumption.
You can check out all four versions here.