When did babylon begin to dominate Mesopotamia

In the 18th Century BC, during the reign of Hammurabi.

Explanation:

During the 2nd millennium B. C. Amorites and Syrians settled down in Mesopotamia, which is again divided into several kingdoms. However, in the 18th century B. C. Hammurabi, king of Babylon, established a unique and stable state that stretched from the Persian Gulf to the city of Mari. City of Babylon became a very important cultural and religious center of Mesopotamia.