Final answer:
The Assyrians primarily battled with the Babylonian Empire for control over Mesopotamia and the Middle East, leading to the eventual fall of the Assyrian Empire and the rise of the Neo-Babylonian Empire.
Step-by-step explanation:
The Assyrians battled with the Babylonian Empire throughout a significant portion of ancient history for control over Mesopotamia and the Middle East. After a period of decline post the Late Bronze Age Collapse and internal strife in Babylonia, Assyria reasserted its dominance during the early stages of the Neo-Assyrian Empire. However, this empire's supremacy was challenged by the growing powers of Babylonia and Media. The final blow came when the Babylonians, allied with the Medes, brought an end to the Assyrian empire around 609 BCE. This fall eventually led to the rise of the Neo-Babylonian Empire, which became the most significant power in the region for several generations following the end of Assyrian control.