The Sumerians were the first people in world history to live in proper cities, and to develop a full writing system. For these reasons they are generally regarded as having the oldest civilization.
Exactly when the Sumerians can be said to have acquired a “civilization” is a moot point (and a somewhat arbitrary one). However, by 3500 BCE they certainly lived in cities several thousand strong, and had almost certainly already developed a simple form of their writing system.
In the late third millennium BCE the Sumerian city states began to lose power to the Akkadians, another people who inhabited Mesopotamia at that time; and from this time onwards the Sumerian language began to give way to Akkadian. However, the Akkadian civilization was so inextricably bound up with that of the Sumerians that it was effectively an extension of it. It was really with the rise of Babylon that Mesopotamian civilization entered a new phase.