Although technically they are "Forbidden One" cards, and not "Exodia" cards, most people know them simply as "the pieces of Exodia". As one the oldest Archetypes in the game, Exodia decks have been around forever.
Exodia Decks don't care about Life Points, because they use an Alternate Win Condition. This allows them to be reckless with their card choices and not have to focus on what the opponent does. Regardless of what you are playing against, the goal remains the same: assemble all five pieces of Exodia and declare yourself the winner. Exodia decks have more draw power than any other type of deck, and are often FTK decks.
Exodia decks typically have no defense, leaving them wide open to opponent's OTKs. They rely a great deal on luck and usually have no way to recover if an Exodia piece is Banished, leaving them with no way to win. They also lack consistency.
I'm going to do this section a little differently from my other guides due to the vast number of Exodia decks that one can make. Deck-specific cards will be listed and discussed in their appropriate sections.
Meet the Archetype
Exodia the Forbidden One Commonly referred to as the "head", this card was one of the first effect monsters, as well as the first to grant an alternate win condition.
This deck uses Royal Magical Library and a ton of draw spells to try and draw out the entire deck in one turn.
Royal Magical Library This is the most important card in Exodia's Library decks. by accumulating 3 Spell Counters on the Library, you can remove them to draw one additional card.
Summoner Monk You can Summon this card and discard a Spell Card to get the Library more consistently. Beware: you can't use Pot of Duality and this card in the same turn.
Magical Citadel of Endymion This card is used to hold additional Spell Counters for the Library. This deck tries to draw out in one turn, and the Citadel is once per turn, so use it and replace it as soon as possible.
Terraforming It's a Spell Card for the Library and it searches the Citadel, thinning the deck.
Pot of Duality Only run one of this, because it can only be used once per turn. Otherwise, it's great for getting to the card you need.
Super Rejuvenation The key card of Dragon's Library, this card lets you draw one for each Dragon you discarded or Tributed that turn. If you draw another copy in the End Phase, you can play it to draw right away.
Heart of the Underdog The key card in this strategy, it lets you draw out massive portions of your deck in a single turn.
Veil of Darkness It functions similarly to Heart of the Underdog, but requires you to discard the monster instead of revealing it, and the monster has to be DARK. Still, having another Draw Spell adds consistency.
Reload Helps get the Heart to your hand faster, and it's a Quick-Play Spell so you can use it during your Draw Phase if the Heart gets interrupted.
Future Fusion Rarely do you see a single card with the ability to thin your deck by so much. Since Zero and Chimeratech have unlimited number of Fusion Material, this card can let you thin your deck by 30+ cards in one turn.
There are a number of Draw loops that can be done. For a near-complete listing, see Exodia OTK. There are too many to list here, so for an idea of how to build a Loop Deck, check out BobaFett2's Loop Guide.