Exodia Decks have changed vastly in Advanced Format. With the release of Emissary of the Afterlife and Dark Factory of Mass Production, they got a little more support with searching and reviving. This type of Exodia deck is my personal favorite; thin the deck out enough so you can just draw and stall.
The premise is self-explanitory. Thunder Dragons, Nimble Momongas, and Giant Rat are the best deck thinners out there currently, so they thin the deck enough that you can use the mass drawing cards to draw Exodia fast. I happen to like Des Lacooda, which is why I have it listed, though some people prefer mass thinning.
There's nothing fancy about this deck. You stall and thin until you draw Exodia out. Simple and effective.
There are other forms of Exodia Decks that are much more ornate. Underdog Exodia uses Heart of the Underdog and a LOT of Normal monsters to (hopefully) draw out the entire deck in a few turns. Usually the only Magic cards are Graceful Charity and three copies of Heart of the Underdog - otherwise, it's all Normal Monsters and the Exodia pieces.
Manticore Exodia is seeing much less play nowadays. The basic strategy behind that is Manticore of Darkness and Card of Safe Return. Manticore of Darkness can be returned to the field at the end of the turn when it's destroyed by sacrificing a Beast/Beast Warrior Monster from your hand or field. By sacrificing a Manticore yourself, you can continuously draw cards with Card of Safe Return, and you simply continue to sacrifice the Manticores until you draw Exodia.
There are other forms of Exodia - people are continuously looking for new ways to thin the deck and get Exodia on turn one. These are the top three forms of Exodia deck, as far as I know. Though Exodia is outdated, it's almost always a surprise, and you may do well at a local tournament by running an Exodia deck.
Monsters - 21 (3 Tribute, 18 Non-Sacrifice)
Magic/Spells - 14
Traps - 7
