In the Anglo-Saxon epic poem, Beowulf, the protagonist Beowulf insists on battling Grendel with his bare hands, instead of using a weapon, because, he says, Grendel doesn't use one.
The sense of honor Beowulf displays here is one of the main themes of the poem. Beowulf sees using a weapon against a beast that uses no weapon as dishonorable. He will fight Grendel on equal terms. It is the honorable thing to do. Grendel is charmed and cannot be cut by swords, but Beowulf does not know this as far as a reader can tell, and this is not why, according to his own words, he refuses to use a sword.
As he says:
...I have heard,
Too, that the monster's scorn of men
Is so great that he needs no weapons and fears none.
Nor will I. My lord Higlac
Might think less of me if I let my sword
Go where my feet were afraid to, if I hid
Behind some broad linden shield: my hands
Alone shall fight for me, struggle for life
Against the monster. God must decide
Who will be given to death's cold grip. (261-270)
Of course, no one else mentions this, and others have fought Grendel with swords. But this is one of the character traits that makes Beowulf different and special. His sense of honor is even greater than that of the rest of his society. That's why he's an epic hero. He could also give a great speech!