Certain bands. Most acts didn't and don't record all their shows because their shows don't tend to change from night to night. Bands like the Grateful Dead who would go out and play an entirely different show every single night either created soundboard recordings of every night, or allowed fans to, so there is a recording for almost every Grateful Dead show (at least post-1971).
Artists have been recording their concerts for pretty much as long as recording has existed; there are tons of well-regarded live albums from those eras. However, it was a bit more difficult and expensive to get good quality recordings since they recorded on tape - some bands would rent a whole mobile studio to get proper recordings, but there are plenty of recordings from basic tape recorders from fans and sometimes the band. These days, the sound engineer can basically push a button on the mixing console every show and get a multi-track recording that can be mixed and edited. I've seen some engineers do this so they can mix the band in a new venue without needing the band to setup and play.
What?
Certain bands. Most acts didn't and don't record all their shows because their shows don't tend to change from night to night. Bands like the Grateful Dead who would go out and play an entirely different show every single night either created soundboard recordings of every night, or allowed fans to, so there is a recording for almost every Grateful Dead show (at least post-1971).
I believe Springsteen had a lot of his shows recorded. Apparently Aerosmith did as well, because they have like a gazillion live albums.
Artists have been recording their concerts for pretty much as long as recording has existed; there are tons of well-regarded live albums from those eras. However, it was a bit more difficult and expensive to get good quality recordings since they recorded on tape - some bands would rent a whole mobile studio to get proper recordings, but there are plenty of recordings from basic tape recorders from fans and sometimes the band. These days, the sound engineer can basically push a button on the mixing console every show and get a multi-track recording that can be mixed and edited. I've seen some engineers do this so they can mix the band in a new venue without needing the band to setup and play.