A major river of Northern Rhovanion, also referred to as the River Celduin.

The source of the River Running was within the halls of Erebor where it issued from a crack in the rock, flowing in a channel from thence to the Front Gate and across the valley below where, in Thror's time Men had spread up-river and built the town of Dale which was later destroyed by the dragon Smaug.

The River Running had two main tributaries; the Forest River which flowed from the Grey Mountains and joined it at the Long Lake, and the River Redwater (River Carnen) which was sourced in the Iron Mountains. The River Running flowed to the south east, finally emptying into the Sea of Rhun and provided an important trade route to more distant lands. At the time of The Hobbit, the only known permanent habitation along its length was Lake-town, built by the citizens of Esgaroth after the arrival of the  Smaug.

NOTE: Additional information from Cirdaneth
Encyclopedia entry originally written by PotbellyHairyfoot