You don't get to become the dominant political power from the Zagros Mountains to the Nile River without military dominance as well. A lot of debate about Assyrian cruelty stems from royal artwork and inscriptions that could be described as propaganda (such as the famous relief of Ashurbanipal seated in front of the Queen of Elam with her defeated husband's head mounted on a spike behind her. However, that cruelty is not the same as actual military prowess, and Assyria's strength in arms would be hard to deny.

First, it should be noted that I'm specifically referring to the Neo-Assyrian Empire (c. 920-609 BCE). Assyria had two earlier periods of military expansion and subsequent contraction in the Bronze Age, but the Neo-Assyrian period is what most people think of when they think of THE Assyrian Empire. Beginning under King Ashur-Dan II and continuing for about a century, the Assyrians became the first major military force to exploit the general power vacuum in the early Iron Age Near East. While Mesopotamia and Greater Iran aren't typically included in the Late Bronze Age Collapse a series of economic and political disasters that included the collapse of the Hittite Empire and Mycenaean Greece as well as the end of the Indus and Oxus Civilizations and the Aramean migrations of the early Iron Age all resulted in the traditional power players of Mesopotamia receding to their core territories for most of the 10th Century BCE.

Over the course of the 9th Century, Assyria rapidly conquered most of their neighbors, including Babylon, from Cilicia in what is now southwestern Turkey to the modern border of Iraq and Iran on the Persian Gulf. Some of these defeated enemies were forced to pay tribute, often with pro-Assyrian kings or even kings from the Assyrian royal family on their thrones. Others were wholly annexed into Assyrian political control. Even during a period of internal division in the early 8th Century BCE, the Assyrians did not lose any significant territory for long despite the best efforts of their rivals. The Kingdom of Urartu to their north was a constant threat, and Babylon regularly rebelled and had brief windows of independence (usually with support from Elam).

In the mid-8th Century, internal divisions were largely resolved and the conquering continued. Assyrian control stretched down the Levantine Coast and a series of kings initiated invasions of Egypt. Over three generations, these campaigns chipped away at the Nubian kingdom of Kush that ruled Egypt at the time, seized control of Lower Egypt, installed the Libyan Pharaoh Necho I as their vassal, and then supported his son in a final reconquest of Upper Egypt. That son, Psamtik I, is often credited with restoring Egyptian independence, but the reality of it seems more like he simply ruled Egypt at the tail-end of Assyrian power and took the financial opportunity to stop paying tribute. His son, Necho II, still invoked their connection with Assyria to justify an invasion of the Levant when the Assyrians were being conquered by the Babylonians from 612-609.

These Assyrian conquests are documented by a wide variety of sources. Babylonian chronicles, Egyptian records of all sorts, and even the Bible document various Assyrian conquests in their respective regions. Assyrian documents that should not be considered propaganda also refer to their victories. Oracles providing Assyrian kings with advice in the middle of a war and administrative records from merchants and bureaucrats document Assyrian military successes as simple matters of fact. All of these types of sources, including the Assyrian ones, also report Assyrian defeats. They were unable to take Jerusalem. Several invasions of Egypt were either unsuccessful or only partially so. The Babylonians, Medes, Urartians, and Elamites all managed battlefield successes against the Assyrian Empire at its height. However, until the empire began a rapid decline in the 630s BCE, they were limited or temporary successes always followed by more impactful Assyrian victories.