I believe eunuchs played a role in Byzantine society at one point. Narses is a famous Byzantine eunuch general. It's something I've wondered about myself too however, so I couldn't say much more on the role of eunuchs in Byzantine society/court.

I assume you're asking about eunuchs in a politician role? The Ottoman empire used eunuchs extensively, and there are some examples of eunuchs holding political power in Europe, even through the 18th century. Eunuchs were around in the ancient world, including Assyrians, the Romans and well through the Byzantine empire. The Hijra of India are also comparable, some are castrated, definitely a societal third-gender. About the only societies that didn't "do" eunuchs were in North and South America, though there are examples of third-gender people in those societies that are comparable.

As it is this question is so broad as a moderator I feel kinda morally obligated to remove it! Can you narrow down what made you ask the question?

Could you clarify what you mean by "non-Chinese society"? Specifically, is Korea included?

The Byzantines had Eunuchs all over the place, often in administrative role and had a completely different hierarchy to the "complete" civil servants. They even had a full choir of over 100 Eunuchs, the seeds of this date back to Aelia Eudoxia in the fifth century) to sing for the emperor and any foreign dignitaries. A visiting bishop, I can't recall his name at the moment, thought that they sounded like Angels.

Eunuchs, or castrati, were very prominent in operas such as this one by Monteverdi, sung by counter-tenor Philippe Jaroussky. Handel and other composers of Italian opera also wrote extensively for castrati The point of it was to castrate boys before they reached puberty and thus they would be able to reach very high notes into adulthood. Boys from poor families would often be castrated because it offered them an escape from poverty and a chance at fame. As famous singers castrati were often society figures and competed with the sopranos to see who could be the greatest prima donnas.

Eunuchs played similar roles to their Chinese counterparts in the Achamenid Persian Empire, rising to highpolitical positions like that of vizier. For example, one eunuch, Bagoas, was vizier of the empire when they reconquered Egypt. He attempted to remain in power by attempting to poison various members of the royal family and install his chosen successors before eventually being executed by Darius II. The instability this caused was one factor that weakened Darius's control and contributed in part to Alexander's ability to conquer the empire shortly after.

He is not to be confused with a different eunuch Bagoas who was a courtier (and possibly lover) of Alexander during his reign/conquests, another role that eunuchs played in Grrek and Roman society.