Goreu was born into a family of giants and tragedy. His father was Custennin, the son of Mynwyedig, a shepherd of enormous size who tended the vast flocks of sheep that belonged to Ysbaddaden Bencawr, the Chief Giant whose fortress was the greatest in the world. Custennin's wife was a daughter of Amlawdd Wledig, the king whose numerous children included Eigyr, the mother of Arthur, and Goleuddydd, the mother of Culhwch. This royal and giant blood made Goreu cousin to Arthur and Culhwch, but it also made him a target. Ysbaddaden Bencawr had murdered twenty-three of Custennin's twenty-four sons, leaving only the youngest alive. Goreu's mother, in her desperation, hid him in a chest at the end of the hearth, keeping him from the giant's sight and hoping against hope that he might survive where all his brothers had perished.

The boy's salvation came with the arrival of Culhwch ap Cilydd, Arthur's cousin, who came to the plain before Ysbaddaden's fortress seeking Olwen, the giant's daughter. Culhwch was accompanied by six of Arthur's most gifted warriors: Cai, the knight with magical qualities; Bedwyr, who never feared any quest; Cynddelig Gyfarwydd, the guide; Gwrhyr Gwalstawd Ieithoedd, who knew all languages; Gwalchmai, Arthur's nephew who never returned home without success; and Menw, who could cast spells of invisibility. They met Custennin tending his enormous flock, and the shepherd told them of his losses and of the boy hidden in the chest. Cai, moved by the boy's plight, promised that if the youth kept company with him, he would not be slain unless Cai himself was slain. The boy emerged from the chest, a youth with curly yellow hair, and joined the quest.

The adventure that made Goreu famous took place at the fortress of Wrnach Gawr, the last of the forty impossible tasks set by Ysbaddaden. Wrnach's caer was built of stone and mortar, the greatest fort in the world, with three baileys that no uninvited guest could cross. Cai gained entry by claiming to be a swordsmith, and Bedwyr followed as a craftsman, but the rest of Arthur's men had no permission to enter. It was Goreu who led them across the three walls, passing through the fortifications as if they were nothing, his companions sticking close to him until they were inside the caer. The warriors, astonished by his effortless entry, declared him the best of all men, and from that moment he was known as Goreu, the Best, son of Custennin. They slew the lodging keepers in secret and then killed Wrnach Gawr himself.

Goreu continued with Arthur's men through the remaining quests, including the great hunt of the Twrch Trwyth, the monstrous boar that was pursued across South Wales and Cornwall. When all forty tasks were completed and Culhwch returned to claim Olwen, Ysbaddaden Bencawr knew his doom was upon him. The giant had prophesied that he would die when Olwen wed, and the prophecy was fulfilled not by Culhwch but by Goreu. The Best laid hold of Ysbaddaden by the hair of his head, dragged him behind him to the mound, and cut off his head with a single stroke. He placed the head on a stake on the rampart of the citadel, and he overcame the caer and the territory, avenging the twenty-three brothers who had died at the giant's hands and claiming his birthright as the last surviving son of Custennin.

Goreu's fame extended beyond the tale of Culhwch and Olwen. In the Welsh Triads, he is named as the one who rescued Arthur from three enchanted otherworldly prisons: three nights in Caer Oeth and Anoeth, three nights imprisoned by Gwen Pendragon, and three nights beneath the Stone of Echeifyeint. In the romance Geraint fab Erbin, he appears as one of the four squires of Arthur's court, riding alongside Geraint on his journey to Cornwall. In Breuddwyd Rhonabwy, the Dream of Rhonabwy, he is listed among Arthur's twenty-four chief councillors. The name Goreu, meaning Best, may have originated from an older form Gorneu, meaning of Cornwall, suggesting a connection to the Dumnonian dynasty of the southwest, and some genealogies make his father Custennin identical with Constantine of Dumnonia, the historical figure who appears in Gildas's history of Britain.