The Hellenistic World ("Hellenistic" from the Greek word Hellas for Greece) is the known world after the conquests of Alexander the Great and corresponds roughly with the Hellenistic Period of ancient Greece, from 323 BCE (Alexander’s death) to the annexation of Greece by Rome in 148/6 BCE (although Rome’s rule ended Greek independence and autonomy it did nothing to significantly change nor did it in any way halt the Hellenization of the world of the day).
Alexander the Great (356-323 BCE) of Macedon first follwed in his father’s (King Phillip II) footsteps in subduing the city-states of Greece and then lead his army on a series of campaigns which successfully conquered the then-known world from Macedon, through Greece, down to Egypt, across Persia, to India. Alexander’s tutor was the Greek philosopher Aristotle (384-322 BCE) and, as Alexander traveled, he spread Greek thought and culture in his wake, thus "hellenizing" (to make `Greek’ in culture and civilizaion) those he conquered.
After Alexander’s death his Empire was divided among his four generals (known in Latin as the Diadochi, the name by which they are still referenced, from the Greek, Diadokhoi, meaning "successors"): Lysimachus, Cassander, Ptolemy and Seleucus. Lysimachus took Thrace and much of Asia Minor; Cassander, Macedonia and Greece; Ptolemy seized Egypt, Palestine, Cilicia, Petra, and Cyprus (thus beginnng the Ptolemaic Dynasty in Egypt which lasted until the death of Cleopatra VII in 31 BCE) while Seleucus took control of the rest of Asia (so founding the Seleucid Empire which was comprised of Syria, Babylon, Persia, and India).
Hellenic influence continued to spread throughout the lands ruled by the Diadochi and Greek dedications, statues, architecture and inscriptions have been found in abundance in every locale. Greek language introduced Greek literature into the former Persian Empire, thereby influencing the philosophical thought and writing of the region (and the same held true for the area known as Palestine where Greek literature found its way into the religious thought and scripture of Judaism). The Great Library at Alexandria, Egypt, which is said to have been started by Ptolemy I, became the most important center for learning in the ancient world. Greek theatre flourished throughout the lands conquered by Alexander and the amphitheaters built during the Hellenistic Period show markedly Greek features no matter the nationality of the architect nor the country of construction (one example being, Ai-Khanoum on the edge of Bactria, modern day Afghanistan).
“The spread of Greek influence and language is also shown through coinage. Portraits became more realistic, and the obverse of the coin was often used to display a propaganda image, commemorating an event or displaying the image of a favored god. The use of Greek-style portraits and Greek language continued into the Parthian period, even as Greek as a language was in decline”(gnostic files).
Even after the rise of the Republic of Rome and then the Roman Empire, Greek language, attitudes, philosophy, understanding and overall culture spread from the civilizations conquered by Alexander the Great and his Generals to others in the East and then north to Europe through trade and, further, by Roman conquest, thereby Hellenizing the entire world of antiquity and influencing virtually every culture of the earth today.