Baugi is the brother of the Jotunn, Suttung.
Odin worked for Baugi disguised as Bölverk to obtain the mead of Kvasir in Suttung's possesion. Bölverk then spent the night at Baugi's place. Baugi complained that business did not go well since his thralls had killed each other (which was due to Odin, unbeknownst to him) and he could not get anybody to stand in for them. Bölverk proposed to do their work in exchange for a draught of Suttung's mead. Baugi agreed, saying that he would try to persuade his brother. During summer, Bölverk did the work as agreed and, in winter, asked Baugi for what he owed him.
But when they both went to Suttung's, he refused to give a single drop of the beverage. Bölverk then suggested Baugi to use a trick. He gave him the drill, Rati, and asked him to dig into Hnitbjörg mountain. After Baugi tried to deceive him, a hole was actually dug and Bölverk slipped into it, having taken the form of a snake. Baugi tried in vain to hit him with the drill.
He was then visited by Gunlod and spent three nights with her. Thus, he could have three draughts of mead, but each emptied a container. He then transformed into an eagle and flew away. When Suttung discovered the theft, he took the shape of an eagle and pursued Odin. When the Aesir saw him, they displaced containers in which he spat his loot out. But Suttung was so close to him that he let some drop backwards. Anybody could drink this part, which is known as the "rhymester's share" (skáldfífla hlutr). But the mead of poetry was given by Odin to the gods and to the men gifted in poetry.
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