When Persian and Median (Kurdish) tribes united and formed the first Persian empire under Cyrus' rule (Achaemenid Dynasty), the first foreign territory they captured was Mesopotamia. That was also the end of Mesopotamia as the culture became absorbed into Persian culture in the following 1000 years or so of Iranian rule.
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The Babylonians were to Iranians as the Greeks were to the Romans. Both the Romans and ancient Iranians absorbed another culture and expanded on it. In the case of Iran and the Babylonians, the Babylonians and Assyrian cultures pretty much got replaced completely. Today, a lot of Assyrians hate Persians for what happened back then and they blame us for their demise, and they are partly right. We were the cause of Mesopotamia's downfall and the subsequent end of Babylonian culture. However, we did absorb a lot from them, and I mean a lot.
Take Ahura Mazda, the chief deity of the Zoroastrian religion and a symbol for Iran and Iranians (most Iranian wear a pendant of Ahura Mazda around their necks, including muslims, and they're sold in every jeweler shop around Iran).
Now this is Assur of Assyria
Perspolis, the capital of Persian culture and identity:
See the resemblance? Basically we took a lot from the Babylonians and made it our own. In every aspect of life they were a great inspiration to Iranian people.