House Tully of Riverrun is one of the Great Houses of Westeros and the principal noble house of the Riverlands. They have held Riverrun for a thousand years and more, lords at the meeting of the Red Fork and the Tumblestone where the moats run wet of their own accord, but were never themselves kings. The rivers had crowns long before the Tullys (the Mudd kings of old, the Brackens and the Blackwoods, the Justmans and the Teagues), and in the centuries before the Conquest the riverlords answered to ironborn masters, for King Harren the Black raised black Harrenhal upon the shore of Gods Eye and ruled the Trident from its walls.
When Aegon Targaryen crossed the Blackwater and Harren burned in his halls, Edmyn Tully was among the first riverlords to renounce the Hoares and ride to the dragon's banner. For that loyalty Aegon raised the Tullys above their fellows: Lord of Riverrun, Lord Paramount of the Trident, and overlord to every Frey, Mallister, Bracken, Blackwood, and Mooton between the Neck and the Blackwater Rush. The riverlords have answered to Riverrun ever since, though no Tully has ever forgotten that many of his bannermen were lords and kings before he was.
By the events of A Game of Thrones Lord Hoster Tully sits the high seat at Riverrun, an old man now and ailing, but in his prime he fought beside Eddard Stark and Robert Baratheon to topple the Targaryens. Both his daughters were promised to the cause (Catelyn first to Brandon Stark and, after his murder, to his brother Eddard, and Lysa to Jon Arryn), so that through them Riverrun is bound by blood to Winterfell and the Eyrie. His son and heir, Ser Edmure, keeps the river marches in his stead; his younger brother Ser Brynden, called the Blackfish for the dark mood he took up after refusing the bride his lord brother chose for him, has long served their niece Lysa as Knight of the Bloody Gate. The Tullys' words are Family, Duty, Honor; their sigil a leaping silver trout on rippling blue and red, a house of the rivers, and never of the sword.