This river is usually called the 'River Running', but from time to time, particulary in The Hobbit, it is referred to as the 'Running River'. The two terms are interchangeable.
Also called the Celduin, the Running was a great river that formed the eastern boundary of Rhovanion. It rose beneath Erebor, the Lonely Mountain, and flowed southeastwards for many hundreds of miles to its outflow into the Sea of Rhûn.