Dothraki are expert riders and their horses are of prime importance in their culture, used for food, transportation, raw materials, warfare, and establishing social Hierarchy.
The Dothraki were based off preexisting people whose cultures were heavily influenced by Horses, such as the Mongolian, Sioux and other horse influenced cultures.
The dothraki warriors would fight in saddle using their curved swords called ‘Arokhs’, Bull-whips or standing on their horses and loosing arrows. 
i could do some really cool animations of the dothraki shooting arrows from horse back.
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