"Yezdigerd, king of Turan, was the mightiest monarch in the world. In his palace in the great port city of Aghrapur was heaped the plunder of empires. His fleets of purple-sailed war galleys had made Vilayet an Hyrkanian lake. The dark-skinned people of Zamora paid him tribute, as did the eastern provinces of Koth. The Shemites bowed to his rule as far west as Shushan. (...) His gilt-helmeted swordsmen had trampled hosts under their horses' hoofs, and walled cities went up in flames at his command. In the glutted slave markets of Aghrapur, Sultanapur, Khawarizm, Shapur and Khorusun, women were sold for three small silver coins -- blond Brythunians, tawny Stygians, dark-haired Zamorians, ebon Kushites, olive-skinned Shemites." -- Robert E. Howard: "The Devil In Iron"
Also called the Sunrise Empire, Turan is a nation with lands first extending only along the western reaches of the Vilayet Sea, but eventually stretching westward over the steppes and desert, reaching the border of Zamora by the time of Conan's reign.
For the early history of the people who dominated Turan, see Hyrkania.
Turan was the wealthiest realm west of Khitai, with the possible exception of Vendhya. It had many ports along the Eastern and western shores of the Vilayet and was mistress of the inland sea. Its capital, Aghrapur, had a huge, magnificent palace called the Sunrise Court.
The ambitious Turanians (often calling themselves Hyrkanians, after their ancient race) made forays in all directions as they attempted to enlarge their empire. They had usurped most of the important caravan cities of the Eastern Desert by Conan's time, crowded the eastern frontier of Zamora, taken over the caravan route to the Far East with the assistance of their kinsmen, the Hyrkanians of the far eastern steppe, and even invaded Vendhya. Their strongest monarch, Yezdigerd, exacted tribute from eastern Shem and Brythunia and fought successful battles with the armies of Hyperborea and Stygia.
After the reign of Conan, the Hyrkanians of Turan took advantage of the disintegrating Aquilonian Empire to gobble up Zamora, Hyperborea, Brythunia, and Corinthia. In the South, they clashed with the Pictish invaders in Ophir, subjugated Shem completely, and overran Stygia. They had seriously overextended themselves by the time they marched into Amazon and had to withdraw to the north, settling down to a long series of wars against the Picts. Then came the ice, and Turan tottered as tides of northern barbarians and Cimmerians wrote finis to one of the greatest empires of the ancient world.
References: Hyborian Age I-II, Thing in the Crypt, Hall of the Dead, Hand of Nergal, Bloodstained God, Witch Shall Be Born, Road of Eagles, Devil in Iron, Queen of the Black Coast, Shadows in the Moonlight, Black Colossus, People of the Black Circle, Black Tears, Flame Knife, Conan the Renegade, Conan and the Manhunters, Shadows in Zamboula, Return of Conan, Conan of the Isles, et al.