Until the 19th century, virtually everything that was known about the Assyrian and Babylonian societies was known through the stories contained in the Hebrew Bible and various ancient Greek texts, especially the histories of Herodotus. In the late 19th century, the cuneiform script was deciphered yielding up the secrets, first of the Akkadian language, and later of the Sumerian language. With the ability to read these languages, scholars were able to drastically and profoundly elaborate our understanding of the Assyrian and Babylonian histories by reading the multitude of documents that were being excavated throughout the region of the modern Middle East, especially from the cities of Iraq.

The primary sources for the modern reconstruction of ancient Assyrian and Babylonian histories come from royal inscriptions, king-lists, and later historiographic texts that were written in antiquity in order to understand the even older, ancient past. These documents, in conjunction with the religious, literary, scholastic, and economic texts, are used to reconstruct a critical history of the societies of Mesopotamia. These texts are as reliable or “trust worthy” as any ancient, primary source can be, but they have a narrower focus than the later Greek and Roman texts, dealing almost exclusively with the court of Assyria or Babylonia (and rarely both).

There are numerous works available that introduce and summarize the most important Assyrian and Babylonian historical texts, beginning with the Sumerian King List, all the way down to the last Chronicles and Greco-Babylonian texts.

Check out our book list for a general list of introductory texts.

http://www.reddit.com/r/AskHistorians/wiki/books/middleeast#wiki_ancient_near_east

If you simply want to see some Assyro-Babylonian “histories” check out Finkel and Van Der Spek's catalog here:

http://www.livius.org/cg-cm/chronicles/chron00.html