Let me wikipedia that for you. There are some reference at the end of the article that I would check instead of relying on wikipedia. The Battle of Gaugamela maybe a better reference with lots of references given.
Xenophon, in Cyropaedia (VI.2.17) states "[...] that scythes of steel have been fitted to the axles, and that it is the intention to drive these into the ranks of the enemy?" -- emphasis mine and source here with more details and other sources.
The Greeks did use chariots (as Homer tells us) but those were mobile platforms to cast spears and move about the battle field. I do not recall Homer describing blades attached to wheels at any point. After that, infantry was much the Greek thing with phalanxes.
I know of no evidence that the Romans used war chariots with or without scythes but that does not meant that they could not use them in games. One of the circus game was said to re-create a naval battle in the Colosseum so why not have a few scythes chariots for those extra fun games.
Chariot races did not feature armed chariots as far as I know, although I maybe wrong. Constantinople maybe a better source to look at but again, I do not recall any armed chariot races there.
Da Vinci had an idea for a scythe chariot as well, see image below. Clearly this was a new design based on a old idea.
PS: I could not find any images of primary sources of scythe chariots. Those might well exists, I just have not found any.