Final answer:
The period when Greek culture spread extensively to non-Greek territories after the conquests of Alexander the Great is known as the Hellenistic period, lasting from 323 to 31 BCE.
Step-by-step explanation:
The Hellenistic Period
The period when Greek culture spread to conquered non-Greeks is known as the Hellenistic period. This era began after Alexander the Great, the king of Macedon, conquered the Persian Empire. When Alexander died in 323 BCE, his empire was divided among his generals, leading to the rise of several Greek-dominated states. These states fostered a rich blend of Greek and local cultures, characteristics of the Hellenistic civilization that lasted until the rise of the Roman Empire in 31 BCE. The Hellenistic period was marked by a prodigious spread of Greek culture and language, influencing a vast region from the Mediterranean to the Near East.