First recorded (as a demo titled “As Time Goes By”) by Mick Jagger & Keith Richards (1964).
Hit versions by Marianne Faithful (US #22/UK #9 1965), The Rolling Stones (US #6/MOR #10 1965).
From the wiki: “‘As Tears Go By’ was one of the first original compositions by Mick Jagger and Keith Richards. Up until that point The Rolling Stones had chiefly been performing Blues standards. A story surrounding the song’s genesis has it that Rolling Stones manager Andrew Loog Oldham locked Jagger and Richards in a kitchen in order to force them to write a song together, even suggesting what type of song he wanted: ‘I want a song with brick walls all around it, high windows and no sex.’
“The result was initially named ‘As Time Goes By’, the title of the song Dooley Wilson sings in the film Casablanca. It was Oldham who replaced ‘Time’ with ‘Tears’. According to Jagger biographer Philip Norman, the song was mainly created by Jagger, in co-operation with session guitarist Big Jim Sullivan (who plays the 12-string guitar on the demo).
“Oldham subsequently gave the ballad (a format that the Stones were not yet known for) to Marianne Faithfull, then 17, for her to record as a B-side. Without even asking if she could sing, Andrew asked her if she wanted to cut the record. The success of the recording caused the record company, Decca, to switch the song to an A-side, where it became a very popular single on both sides of the Atlantic Ocean.”
Marianne Faithful, “As Tears Go By” TV performance (1965):
The Rolling Stones, “As Tears Go By” original mono single release (1965):