All civilizations depend on available water, but not all depend on rivers. Rivers also provided ancient societies with access to trade -- not only of products, but ideas, including language, writing, and technology. River-based irrigation permitted communities to specialize and develop, even in areas lacking adequate rainfall. For those cultures that depended on them, rivers were the life blood.
In "The Early Bronze Age in the Southern Levant," in Near Eastern Archaeology, Suzanne Richards calls ancient societies based on rivers, primary or core, and non-riverine (e.g., Palestine), secondary. You'll see that the societies connected with these essential rivers all qualify as core ancient civilizations.
Mesopotamia was the area between the two rivers, the Tigris and the Euphrates. The Euphrates is described as the southernmost of the two rivers, but also appears on maps to the west of the Tigris. It starts in eastern Turkey, flows through Syria and into Mesopotamia (Iraq) before joining the Tigris to flow into the Persian Gulf.
Whether you call it the River Nile, Neilus, or Egypt's River, the Nile River, located in Africa, is considered the world's longest river. The Nile floods annually because of rains in Ethiopia. Beginning near Lake Victoria, the Nile empties into the Mediterranean at the Nile Delta.
Saraswati is the name of a holy river named in the Rig Veda that dried up in the Rajasthani desert. It was in the Punjab. It is also the name of a Hindu goddess.
The Sindhu is one of the rivers sacred to Hindus. Fed by the snow of the Himalayas, it flows from Tibet, is joined by Punjab rivers, and flows into the Arabian sea from its delta south-southeast of Karachi.
The Tiber River is the river along which Rome formed. The Tiber runs from the Apennine Mountains to the Tyrrhenian Sea near Ostia.
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The Tigris is the more easterly of the two rivers that defined Mesopotamia, the other being the Euphrates. Starting in the mountains of eastern turkey, it runs through Iraq to join up with the Euphrates and flow into the Persian Gulf.
The Huang He (Huang Ho) or Yellow River in north central China gets its name from the color of silt flowing into it. It is called the cradle of Chinese civilization. The yellow River is the second longest river in China, second to the Yangtzi.