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It is correct to say that ancient civilizations, including the Mesopotamian empires of Sumer and Babylon, have influenced modern western society in many different aspects.
Let's not forget the cultural influence of ancient Egypt, and India as well
However, if we have to pick one ancient civilization that has had the greatest impact on the Western world, probably it has to be the ancients one: Sumeria.
And we say it was Sumeria because it represented the beginning of human civilization according to historians. Indeed, Sumeria was the oldest civilization on planet earth.
One of the most important influences on other cultures was the development of powerful city-states such as Eridu, the first human settlement on earth, and other city-states such as Ur, Uruk, Lahasg, Nippus, and Kish.
And Sumerians also heavily influence other cultures and civilizations such as the Greeks and later the Romans through their interesting pantheon of gods. Yes, many deities that ruled Sumeria and this symbolism passed through the years to other pantheons, including the Olympus Gods of the Greeks.