Are Babylonians, Sumerians and Mesopotamians the same?

As pointed out by Balaji Viswanathan, Greeks called the modern-day Iraq as Mesopotamia and its people Mesopotamians.  Not only Iraq but Mesopotamia was a term for the area of the Tigris-Euphrates river system covering Kuwait, the northeastern part of Syria and to a much lesser extent the southeastern parts of Turkey. Mesopotamia included Sumer, Akkadian, Babylonian and Assyrian empires, all native to Mesopotamia. Now, what is the difference between Sumerians and Babylonians?
Sumer was one of the ancient civilization in southern Mesopotamia during the Chalcolithic and Early Bronze age.
Babylonian was an ancient post-Sumer civilization that originated in the central-southern region of Mesopotamia. Sumerian spoke the Sumerian language while the Babylonian spoke the Akkadian language despite, its Amorite founders and Kassite successors were not native to Akkadian.

Pre-Babylonian Sumero-Akkadian era in Mesopotamia

Mesopotamia is a Greek word that means the land between the rivers. It is used by the Greeks to describe the cultures between the Tigris and Euphrates rivers. It was not used by the locals themselves.

The people of that land originally referred to themselves as ùĝ saĝ gíg-ga and their land as  kiengi. However, for various reasons their rival group called the Akkadians called this region Sumer. Sumer was the oldest period in Iraqi civilization about 5000 years ago.

Around 3800 years ago, the western Akkadians eventually took over the land of the Sumerians. There they established the next civilization called Babylonia and the name derives from the Akkadian word Babili meaning the gates of gods. The Greeks then corrupted it to term Babylonia.

The present name for the region, Iraq derives from the famous Sumerian city of Uruk that was later used by the Arabs to call the whole region.

In summary, Sumer is the first of those civilizations and named that way by the Akkadians. Later, the Akkadians took over Sumer and established Babylonian civilization. Much later, the Greeks came and called the region Mesopotamia - the land of two rivers. None of these words were coined by the people of the region themselves. The modern word Iraq thus is used as it is a word Sumerians used for one of their important cities of Uruq.

No, they are different.

Sumer was the first great culture to arise in Mesopotamia, arising around the cities of Uruk and Eridu in beginning around 5500 BCE. It was a culture of city states in the lower part of the region between the Tigris and Euphrates rivers. The rise of the city of Akkad, which created an empire that dominated Mesopotamia.

Babylon appeared later, around 2500 BCE, after being founded during the Akkadian period, which followed the decline of Sumer. It became a great city, for a time the largest in the world, and twice was the capital of an empire. The first is often called the (Old or First) Babylonian Empire, the second is generally called the Chaldean Empire. It declined after the Persian conquest of the region.

EDIT: Since the original question has been merged with one that asks about Mesopotamia, I will add a brief response. Mesopotamia is a term derived from the Greek for "land between rivers." It is a geographic expression referring to the area which includes the drainage basins of the Tigris and Euphrates rivers, most of which is in modern-day Iraq. Mesopotamia was, like the basins of the Nile, Indus, and Yellow rivers, one of the first sites of human civilization. A series of cultures, beginning with that of Sumer, arose in Mesopotamia. The early cultures appeared near the mouths of the two rivers, which are today conjoined, but in ancient times were separate. Later cultures, like the Babylonian and the Assyrian, were centered further up the region.

No. Mesopotamia is the region between the Tigris and Euphrates Rivers and all the various peoples who resided in that region are collectively known as "Mesopotamians." The Sumerians were the earliest literate people to inhabit the region of Mesopotamia, although they were not the first people to inhabit the region period, since they were preceded by the Ubaidians, an earlier illiterate culture about whom very little is known. The Sumerians lived in city-states in the southern part of Mesopotamia, across the southeast part of what is now Iraq during the late fourth, third, and early second millennia BC.

The largest and most prominent Sumerian city-states were Unug (or "Uruk"), Nippur, Urim (or "Ur"), Eridug, Sippar, Larsa, Isin, Kish, Larsa, Zabalam, Adab, Girsu, Bad-tibira, Shuruppak, and Lagash. Each of these cities had its own government and its own culture. In around 2334 BC, the Sumerians were conquered by the Akkadian Empire under Sargon of Akkad, which was the first true empire in Mesopotamia. The Akkadian Empire eventually collapsed in around 2154 BC.

The collapse of the Akkadian Empire was followed by a resurgence of Sumerian culture that has sometimes been called the "Sumerian Renaissance." The rulers of the Third Dynasty of Urim (c. 2112–c. 2004 BC) established great prominence during this period and the city-state of Lagash flourished under the rule of King Gudea (c. 2144–2124 BC). Urim was sacked by the Elamites in around 1940 BC during the reign of Ibbi-Sin. This brought an end to the prominence of Urim and the Sumerian Renaissance. The Sumerians gradually assimilated into the East Semitic cultures around them over the course of the next few centuries, but the Sumerian language continued to be used as a liturgical language, similar to how, in much later times in Europe, Latin continued to be used for liturgy even after it ceased to be commonly spoken.

The Babylonians were a later East Semitic people who resided in the same region of Mesopotamia as the Sumerians. Unlike the Sumerians who lived only in city-states, the Babylonians established several different empires in the region over the course of Mesopotamian history. The Old Babylonian Empire was the first Babylonian Empire and it began in around 1830 BC or thereabouts. The most famous ruler of the Old Babylonian Empire is Hammurabi (c. 1792–1750 BC), who greatly expanded the empire and also issued the Code of Hammurabi, a highly influential law code that greatly impacted all Near Eastern law codes written after it.

The Old Babylonian Empire lasted until around c. 1531 BC when Babylon was sacked by the Hittites and fell under the control of the Kassites, who established a dynasty. This period of Babylonian history is known as the "Kassite Period" because it is when Babylon was ruled by the Kassites. The Kassite Empire fell apart in around until around 1155 BC or thereabouts as part of the Late Bronze Age collapse, in which nearly every civilization in the Near East suddenly all fell apart at around the same time, probably due to a variety of causes including draught, famine, roving invaders, government and social collapse, disease, and natural disasters.

There were several centuries of low material culture following this collapse, for which we have very few records and our knowledge is severely limited. During the Early Iron Age, the Babylonians were ruled by the Neo-Assyrian Empire (911–609 BC), which was widely detested by those it conquered due to its particularly harsh policies. As the Neo-Assyrian Empire was falling apart in the late seventh century BC, the Babylonians established the Neo-Babylonian Empire, which lasted from around 626 BC until 539 BC, when it was conquered by the Persian Achaemenid Empire, ruled by Cyrus the Great.

The Neo-Babylonian Empire is the most famous and best-recorded of the Babylonian Empires and it is the one that conquered Judah and brought the Judahites into captivity in Babylon. The most famous king of this empire was probably Nebuchadnezzar II, who is alleged to have built the Hanging Gardens of Babylon, although we cannot actually be sure that the so-called "Hanging Gardens" actually existed, since some scholars think they may be a product of later Greek imagination.

This is a very, very basic introduction to who the Sumerians and Babylonians were and there is so much more that I really ought to have included here but have not. I strongly recommend doing some more reading on the subject because it really is truly fascinating. A very accessible, but still informative (albeit mildly outdated) book on the Sumerians is The Sumerians: Their History, Culture, and Character by Samuel Noah Kramer, originally published in 1963 by the University of Chicago Press. Kramer was the leading scholar on the ancient Sumerians for most of the twentieth-century. Although he was quite an avowed Sumerophile and his book can be rather heavy-handed at times when praising them, he really did know them very well. We have learned more about the Sumerians since the book was published, but, unfortunately, I am not aware of any comparable books about them that have been written more recently.

Akkadian was already a Semitic language related to Hebrew and Arabic. Subsequent languages like Aramaic were too.

Sumerian is the earliest written language (followed closely by Egyptian) and not visibly related to any other language, but afterwards the region had Semitic speech all the way to the present.

They aren't.

The Sumerians were a people who lived in southeastern Mesopotamia from about the late fourth millennium B.C. onwards. Sumerian is a language isolate, i.e. is not related to any other known language.

The Assyrians and Babylonians were two closely related Semitic peoples who settled in Mesopotamia later on - it is common enough to view the Assyrian and Babylonian languages as dialects of what is called Akkadian. The Assyrians lived in what is now northwestern Iraq; the Babylonians in what is now central and southeastern Iraq. The Sumerians eventually were absorbed into the Babylonians.

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