there are a total of 7 classical dance forms in India .they are 1]Kathak2]Bharatnatyam3]Manipuri4]Odissi5]Kathakali6]Mohiniattam7]Kuchipudi ....by Namrata Debroy
Across the world there are very many dance forms.
I'm not sure if it's as traditional as ballet, but I know it dates back pretty far...at least to the early 1900's.
kerala
Both odissi and bharatnatyam are the mid-20th century reconstructions, with roots in Natya Shastra.