I’m going to talk about the differences between Asian series, also called Asian dramas, and American series.
I love Asian dramas because I find them more realistic and more funny. If you are a schmaltsy person, you will love them. I find the actors more handsome and actresses more beautilful… Moreover, good music (OST) accompanies dramas.
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Korean Drama
There are two kinds of drama, especially for Korean dramas : a happy end and a tragical or cruel end. They are very famous and successful due to the sad love stories, and also due to the incredible but funny stories. You can see some extract at Youtube. The most popular dramas are “Full House” (happy end), “Sad Love Story”, “Princess Hour” (happy end), “Winter Love song”… However, there is often redundancy between the sad Korean dramas : when somebody is ill, if it’s not a cancer, he is blind.

Full House

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Taiwanese Drama
Their very particular stories with happy end is very attractive. The public is rather for young people (children, teenagers and young adults) than the others. I advise you to see “Devil beside you”, “Heaven’s wedding dress”, “My lucky star”…

Devil Beside You

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There are also Cantonese and Japanese dramas, but I don’t see them very much.