No professional, alas, ever hauled in cameras to record most of the great theatrical events of the previous century. Nowadays, most shows get taped at some point but there's no full, real video of 99% of what has appeared on the great stages of the world. We have to settle for the few snippets that exist here and there.
Here's a great snippet — Julie Andrews performing a number from the original My Fair Lady. This is not a video from an actual performance of the show but it's darned close. Ms. Andrews appeared in My Fair Lady from its opening in March of '56 'til August of '59 with a few extended vacations during that period. In early 1962, while she was still appearing in Camelot with Richard Burton and Robert Goulet, the three of them participated in a TV special, The Broadway of Lerner and Loewe. On it, she performed this number from My Fair Lady, which was still running at a theater a few blocks away. They borrowed the sets and costumes and, I'm told, the actor who was then playing Freddie (sorry I don't know his name) and recorded Julie doing the number exactly as she'd done it on stage a few years earlier. Great stuff.
You'll notice, by the way, this video is in color. I have The Broadway of Lerner and Loewe on DVD but in black-and-white. Anyone out there know where to get a color copy? It's a great special full of gems like this. I'll link you to another one tomorrow if I think of it.