Hello I'm from Taiwan and I speak Chinese :) We say Mid-Autumn Festival instead of Mid-Autumn Day Festival And you could say Moon Festival as well. Happy Mid-Autumn Festival & Happy Moon Festival in Chinese would both be 中秋節快樂 中(zhong) = mid 秋(qiu) = autumn 節(jie) = festival 快樂(kuaile) = happy ps. The reason why we also call it Moon Festival is because of the moon of that night. It's always so bright, clear and full. If you wanna give your friends some gifts with the wishes and you wanna say something formal to them, you could say 中秋佳節愉快! This has the same meaning as the upper one but with the formal aspect. Hopefully these could help ^__^

Abroad have the Mid-Autumn festival?

中秋佳節愉快 ! (金風送爽,) 八月佳節愉快 ! 中秋節 又稱 八月節

zhongqiujie kuaile!

Happy Mid-Autumn Day Festival = 中秋節快樂!