When someone mentions Beatles, Rolling Stones usually pops into minds, and reverse also. When, or did the two hottest bands of the same era hook up? They did meet when the Beatles attended a Rolling Stones concert after they had a couple of hits, and the Stones did not. Bill Wyman mentions in the UCR story, "Halfway through the set we kind of glanced up, and there were four silhouetted leather-clad persons standing just in front of the bandstand in amongst all these kids who were all dancing and playing around," then added The Beatles were "being totally ignored by the audience."
After seeing then in a crowd at their show, they did hang out together, "So when we finished our set, we got chatting to them and had beers, and they stayed for the second set. And afterwards we all went back to the flat in Chelsea, where Mick [Jagger> and Keith [Richards> and Brian [Jones> lived, hung out all night playing blues music, just talking about music and became great mates." Sounds like the Beatles admire them almost as much as I do.