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Which way round is it best to write names from that part of the world?

For example, Im about to pen a lettet to Zhang Ziyi, buy am unsure whether to put Ziyi Zhang instead? Is there a prefered way? In the letter itself should I write To Ziyi or to Zhang?

I actually think that you should use the celenbrity's full name

In Chinese culture, the preferred way to write someone's name is last name then first name. If you are sending the request to China, you should address it:

Zhang Ziyi

But if you aren't sending it there, just address it normally:

Ziyi Zhang

I'm not sure about Japan or Korea.

JC

As said above with Chinese you put last name first. The same goes for Japanese names. Such as Utada Hikaru. Her first name is really Hikaru. But you put last name first so it's Utada Hikaru.
As far as Korean goes I think it's first name first. I had a friend named Zi Hyun Lee and I asked her how I should address a letter to her that I was going to send to Korea and she told me to write her first name first then Hyun Lee. So I suppose it's that way.