"Not Fade Away" is credited to Buddy Holly (originally under his first and middle names, Charles Hardin) and Norman Petty (although Petty's co-writing credit is likely to have been a formality) and first recorded by Holly and his band, the Crickets. Until the end of his career, Holly recorded with his group, The Crickets, but he set up a deal with their record company, Decca Records, to release some songs under his name and have others credited to the group. This was credited to The Crickets and released on the Brunswick subsidiary. Songs credited to Buddy Holly came out on Coral Records. Drummer Jerry Allison played a cardboard box for percussion on this. He'd heard Buddy Knox' drummer do the same on "Party Doll."

The cover by the Rolling Stones was one of the Stones first hits. Recorded in January 1964 and released by Decca Records on February 21, 1964, with "Little by Little" as the B-side, it was their first Top 10 hit in Great Britain, reaching number three. In March 1964, it was also the band's first single released in the United States, on the London Records label with "I Wanna Be Your Man" as the B-side (It had been briefly preceded by "I Wanna Be Your Man" with "Stoned" as the B-side, but this was quickly withdrawn). Their manager, Andrew Oldham, was convinced the Stones would be successful after hearing what they did with this. Said Oldham: "Although it was a Buddy Holly song, I considered it to be like the first song Mick and Keith wrote, in that they picked the concept of applying that Bo Diddley thing to it. The way they arranged it was the beginning of the shaping of them as songwriters. From then on they wrote. At that time, Mick, Keith, and I lived together. They were into the last half bottle of wine and going through, it was one of those magical moments. When Keith played that to me in the front room you could actually HEAR the record in that room. What basically made the record was that whole Bo Diddley acoustic guitar thrust. You heard the whole record in one room. We gotta record it! But there's no way if someone had just said coldly, Right, let's do "Not Fade Away" that we would have wanted to do it without hearing the way that Keith was playing it on the guitar. Keith just did it. And that was that. To me, they wrote the song. It's a pity we couldn't have gotten the money."

Charlie Watts: "We did it with a Bo Diddley beat, which at the time was very avant garde for a white band to be playing Bo Diddley's stuff. It was a very popular rhythm for us in clubs; looking at it from the drumming point of view. So we did it in this slightly different way than Buddy Holly did it." Phil Spector is credited with playing maracas on the record but in fact he was playing an empty cognac bottle with a 50 cent piece.

Bill Wyman: "The rhythm thing was formed basically around the Buddy Holly thing. We brought the rhythm up and emphasized it. Holly had used that Bo Diddley trademark beat on his version, but because he was only using bass, drums and guitar, the rhythm element is sort of a throwaway. Holly played it lightly. We just got into it more and put the Bo Diddley beat up front."

In 2004, this song was ranked number 107 on Rolling Stone's list of "The 500 Greatest Songs of All Time". The Crickets' recording never charted as a single.