On October 1, 331 BC, Alexander of Macedon attacked a Persian army considerably larger than his own at Gaugamela, on a plain east of the Tigris in what is now northern Iraq, and by the end of the day Darius III had run. The Persian king escaped. His army did not. Within weeks Babylon and Susa opened their gates, and the Achaemenid state, which had ruled the Near East for two centuries, had no field army left to put between Alexander and its treasuries.
Darius had done nearly everything right this time. He picked the ground himself and had it levelled so his scythed chariots could run. He assembled contingents from most of the empire. The numbers in the ancient sources are absurd, but modern estimates put his force somewhere between 50,000 and 100,000 against Alexander's roughly 40,000. He had also drawn the Macedonians away from the coast and into Mesopotamia, where they could not be supplied by sea.
Alexander's answer was the oblique advance, and it worked because the two armies were not equally trained. He refused his right wing, drew the Persian left out of line, and drove a wedge of companion cavalry into the gap that opened in front of the king. The Persian centre held. The wings did not. When Darius' own charioteer was killed beside him, the story goes, word ran down the line that the king was dead, and the right wing broke and rode for home.
It broke because the Macedonian line was a professional instrument and much of the Persian one was not. Alexander's phalanx drilled with the sarissa, a pike about 18 feet long, and his cavalry could wheel and charge on command. A great deal of Darius' infantry had been recruited that year. The gap Alexander found was not luck so much as a consequence: a levy army cannot hold formation once its flank goes.
Darius got away to Ecbatana and was killed in 330 BC by Bessus, his own satrap of Bactria, who then claimed the throne for himself. Alexander took Persepolis and burned its palace in the spring of 330. The empire Darius had inherited, the largest the world had yet seen, was divided among Alexander's officers within a decade of his death at Babylon in 323 BC.