* The Assyrians: In the 7th century B.C.E., the Assyrians, under kings like Sennacherib and Ashurbanipal, conquered Babylon and ruled over the region for several decades.
* The Neo-Babylonians: After a period of Assyrian rule, the Babylonians revolted and established the Neo-Babylonian Empire under King Nabopolassar. They conquered the Assyrians in 612 B.C.E. However, the Neo-Babylonian Empire itself was eventually conquered by...
* The Persians: In 539 B.C.E., Cyrus the Great of Persia, founder of the Achaemenid Empire, defeated the Babylonians and brought their empire to an end.
So, depending on which specific period you're interested in, the answer to who conquered the Babylonians in B.C.E. could be either the Assyrians, the Neo-Babylonians (briefly), or the Persians.