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The Nazgûl, a.k.a. the Ringwraiths, a.k.a. the Black Riders, have swords, of course. But they don't really seem to need them: their very bodies are weapons of evil. Their voices are so filled with hate that they "[pierce] the heart with a poisonous despair" (5.4.28). Their breath is apparently so evil that the Witch-king of Angmar manages to send Éowyn, Merry, and Faramir into comas just by breathing on them (that's some serious halitosis). Their very presence on the battlefield freaks people out: when Faramir's troops meet five of the Nazgûl at the fields of Pelennor, they immediately start running away on foot, they are so frightened.
How do you even fight an enemy that literally breathes despair? The answer is: with difficulty. It takes Éowyn and Merry working together to bring down just one of the Nazgûl, and he kills Théoden and injures Faramir and countless others before they get even the chance. (See our "Character Analysis" of Éowyn for more on this battle with the Lord of the Nazgûl.)
The Ringwraiths basically symbolize fear itself: Gandalf describes the Lord of the Nazgûl as "a spear of terror in the hand of Sauron, shadow of despair" (5.4.115). After all, one of Sauron's main weapons is fear, and the Ringwraiths help him spread it around. Since they are so linked to Sauron, it's not until Sauron himself falls that the Ringwraiths die with him.
As for their characters, how much character depth can you really give to creatures who are physical representations of "terror" and "despair"? We find out that the Witch-king of Angmar was evil even during his lifetime, before getting Wraithified (he was the one who burned down the original Weathertop, according to the Appendices that follow The Return of the King). And his second in command has the fabulous name of Gothmog, which would make a great cat name if it didn't seem like bad luck to name your cat after one of the Nazgûl.
Still, we will give them credit for one thing: their deaths sound pretty spectacular. When Sauron finally realizes, just a bit too late, that the hobbits are trying to throw the Ring into the Cracks of Doom right in the middle of Mordor, he calls the Ringwraiths back ASAP. As they fly to their master, they are "caught in the fiery ruin of hill and sky" (6.3.95) as Mount Doom explodes. Whoops.