In 1922 violinist Lev Tseitlin founded an orchestra according to Soviet principles of collective responsibility: it had no conductor.
So apparently in Western classical music, conductors are a relatively recent addition. In the 18th century it was more common for one of the performers to keep the tempo and communicate with the rest of the orchestra through gestures and movement (waggling their violin bow for example).
As we get closer to the 19th century, though, orchestras began getting larger and as a result started employing dedicated conductors to keep everyone playing together. Mozart conducted the 1791 premiere of his opera The Magic Flute, for example, and by the 19th century conductors were the norm rather than the exception. It’s also around this time that conductors began waving batons rather than just using their hands or pieces of rolled up paper – Felix Mendelssohn was an early baton-wielder.
Jumping forward to the 1920s, the Russian Revolution was in full swing and the idea of distributing authority back to the masses came with it. Those ideas weren’t just political, they spread into the arts in some interesting ways. The influential Soviet film The Battleship Potemkin, for example, has no individual hero – instead, the revolutionary sailors act as a kind of collective group protagonist.
The violinist Lev Tseitlin, in 1922, tried something new. Remove the conductor, arrange the orchestra in a circle so that the performers can see each other, and work things out via committee. This was Persimfans, the first conductorless orchestra. I couldn’t find a video of the original online, but here’s a modern revival of Persimfans to give you an idea of how well this worked:
Nowadays there are quite a few conductorless orchestras out there, although they often lean towards smaller orchestral groups and chamber music ensembles.
[Thanks to Frazer E. for suggesting this topic.]
The Generalist Academy 
When I was in high school, the band would play “A Christmas Festival” by Leroy Anderson each year* as part of our concert. We would go to each elementary school and perform. Our director’s shtick was for us to pretend to fall apart when he directed. Repeat a few times with more and more frustration from him. Finally he would choose a little kid to come up and direct us. Of course we played it perfectly for the kid to the joy of the entire school.
Totally different from the topic of this post, but it made me think of those years and smile (which was sorely needed today).
*I listened to it K-6 and played it 9-12. Then listened to it again the next 8 years as my brothers went through high school.