In 539 BC Cyrus the Great of Persia conquered Babylon. The Bible records in the Book of Daniel about the "Handwriting on the Wall" where Belshazzar who had been ruling in Babylon on behalf of his father Nabonidus, saw handwriting on his palace wall during a feast, which Daniel the Hebrew interpreted as the end of the Babylonian Empire.
Alexander and the Greeks
From the time of the first Sumerian cities (c 3500 BC) to the invasion of Mesopotamia by Alexander the Great, King of Macedon (331 BC) there was essentially only one civilization in Mesopotamia although there were many waves of immigrants.
The immigrants during that period did not have any well-structured culture or civilization until they adopted the culture of the city dwellers. Of course these people brought some things with them. The invaders had different languages and more importantly they had a tribal society with tribal customs. However after a few generations living in cities these tribal customs were virtually forgotten. The newcomers also had different views on some aspects, such as war. Sumerian kings boasted of the periods of peace they had brought their kingdoms, whereas the Akkadians, a Semitic people, boasted of their great victories. However there were only minor differences in Mesopotamian civilization until the introduction of Hellenism (Greek culture) with the invasion by Alexander.
The Greeks were a people with a long history of civilization. They imposed this civilization on all the people they conquered, building new cities with Greek civilizations. Babylon was no longer the principal city in the area and began to decline. As Babylon declined so did the Mesopotamian civilization. The old customs were forgotten or not performed. The old gods were abandoned. The old cities, the great cities of Sumer and Akkad, such as Babylon and Ur, Uruk and Eridu, Lagash and Isin declined to insignificance.