Dueling was the art of battling a relatively few number of foes with a mêlée weapon or magic.
Honor duels
People in Alagaësia would sometimes battle each other over personal issues. The most common reason was when one party offended the other. In many instances these duels occurred when one family declared a blood-feud with another.
Wizard duels
A wizards' duel took place when two magicians battled each other with their minds, concentrating on prevention of magic use by the other party.
Magician duels
Magician duels occurred when magicians battled each other without magic, or with limited use of magic.
Known magician duels
Du Fyrn Skulblaka
While no specific duels are mentioned, many battles were fought between elves and dragons, two magical races. These duels only came to an end when Eragon discovered a dragon egg, and together they formed the Dragon Riders.
Vrael vs. Galbatorix
During the Dragon War, Vrael, the last leader of the Dragon Riders, battled the rebel Galbatorix. Galbatorix killed Vrael and took over Alagaësia, establishing the Empire.
Eragon vs. Durza
At the close of the Battle under Farthen Dûr, Eragon dueled with Durza the Shade. Durza wounded Eragon, but thanks to Arya and Saphira, Eragon managed to defeat him, allowing the Varden to take advantage over the fighting Urgals.
Eragon/Saphira vs. Murtagh/Thorn
At the Battle of The Burning Plains, Eragon and Saphira battled Murtagh and his dragon, Thorn, who had been forced to join Galbatorix. Eragon and Saphira were tired from the fighting and Murtagh was able to pin Eragon with magic. In the end, Murtagh chose to let Eragon go.
Brom vs. Morzan
After Brom's dragon, Saphira, was killed, Brom set out on an adventure to destroy the one man he felt most responsible, Morzan. Brom had many battles during his travels, but eventually he ended up dueling Morzan and defeating him. He then took Zar'roc as a token.
Eragon vs. Varaug
During the battle at Feinster, Eragon engages Varaug with his mind. Just when Varaug begins to win, the shade's concentration is broken, and Arya kills the shade.