Some popular draw engines at present are:
Destiny Draw Engine
The Destiny Engine functions by using "Destiny Draw" to discard "Destiny HERO" monsters whose effect(s) are more beneficial from being in the Graveyard, such as "Destiny HERO - Malicious" or "Destiny HERO - Dasher". "Elemental HERO Stratos" is commonly used to get the "Destiny HERO" you need, and so "Reinforcement of the Army" is used. This became quite a useful method of summoning the Monarchs, creating the so-called Perfect Circle Deck.
This deck also works well with "Card of Safe Return".
Allure Draw Engine
This works by banishing DARK monsters, such as "Dark Magician of Chaos", then reviving those monster(s) with either "Dimension Fusion", "Return from the Different Dimension" or more popularly, "Escape from the Dark Dimension". With the March 2010 Lists it was Limited, severely hampering this draw engine.
Card of Safe Return Engine
Can be better known as CoSR Engine. This Draw Engine uses the effect of "Card of Safe Return" to get an extra Draw for each monster special summoned from the grave and get Hand Advantage. This engine is especially useful and really only good in Decks that can Special Summon many monsters from the Graveyard. Zombie Swarm and Dark World Decks are great decks to use with this Draw Engine.
A variation of this draw engine uses 2 "Manticore of Darkness" to continuously loop by sending 1 to the graveyard and sending the second one from your hand or deck to bring out the next one. The "Manticore Loop" is often used in Exodia OTK decks.
Trade-In Draw Engine
Although this may not be a particularly popular draw engine, "Trade-In" has an advantage over other draw engines; it is neither Limited nor Semi-Limited. This draw engine discards powerful Level 8 Monster Cards to the Graveyard, usually to revive later. This can be useful in multiple decks, but is found commonly in Dragon decks (which can also combined with "Super Rejuvenation"), Darklords, and Destiny HEROes. Basically this draw engine is viable to every deck that can utilize many Level 8 monsters in the deck.
This draw engine is used in Lightsworns, but can be splashed into any deck that benefits from sending many cards to the graveyard. It consists of searching out Lightsworn monsters with "Charge of the Light Brigade" and then discarding them with "Solar Recharge". Doing those two things thins your deck by a total of 8 cards, sending 6 of them to the graveyard - a very fast maneuver, in short. The Recharge draw engine can be used in Lightsworns, of course; it is also (as for now) commonly used in Chaos Dragon decks. When this draw engine is used outside of Lightsworn, some of the two most common Lightsworns used are "Ryko, Lightsworn Hunter" and "Lyla, Lightsworn Sorceress". Unlike most other Lightsworns, their effects are non-specific to Lightsworns, and can give the ability to destroy spells and traps to decks that do not normally have it.
Dedicated Lightsworn decks also use the effect of "Garoth, Lightsworn Warrior" to draw many new cards. With its effect, you can thin your deck even more quickly than normal.
"Charge of the Light Brigade" is Limited by the March 2010 Lists, hampering this engine's severity - though not as severely as the Allure and Destiny Draw engines. As a replacement, you can use "Reinforcement of the Army" - though it is not as fast as Charge, and you will have to use Lightsworn monsters like "Jain, Lightsworn Paladin", it is still useful.
Reckless Greed Engine
Using 3 "Reckless Greeds" together lets you draw 6 cards and skip only 2 Draw Phases, since their drawbacks don't stack. "Mask of Darkness", now at 3, can bring back your "Reckless Greeds" and allow you to draw even more. This engine is commonly used in decks that do not have ready access to any other. When combined with other, more consistent draw engines, the extra cards can bring a victory before the negative effects come into play. This draw engine is commonly used in Dark World and Exodia decks.
Heart of the Underdog Engine
"Heart of the Underdog" is a card that allows you to draw extra cards into your Draw Phase by drawing a Normal Monster. The advantage of this engine is when you run Normal Monsters at your Deck, your draw power shall be increased and you may yet draw constantly if you draw a lot of Normal Monsters during your Draw Phase, causing you to speed up your Deck sharply. This engine benefits from a Deck that does not use almost any Effect Monsters, but abuses Special Summon and searching effects like "Emissary of the Afterlife" and "Enchanting Fitting Room". This engine is mostly used for Exodia Decks, but also works pretty well at "Ojama" Decks.
Good Goblin Housekeeping Engine
Good Goblin Housekeeping is a card with a ruling that allows you to draw an extra card if, upon resolution, it is in the graveyard. So, combining this with cards such as Emergency Provisions, Twister, and Mystical Space Typhoon, one can add additional draw power, as well as provide additional benefits, during or outside of the draw. Coupled with cards like Rose Fairy, Watapon and Blackwing - Breeze the Zephyr and additional draw support, special summons can come quite easily for tribute fodder.
Card of Sacrifice Engine
Card of Sacrifice, much like Reckless Greed, is a card that allows you to draw two cards but with less of a negative effect. Decks that do not attack such as Burn Decks, or even some of Naturia decks benefit as they suffer no drawbacks from this card. They can also be stacked to draw multiple cards in one turn since the drawbacks affect only this turn (much like "Reckless Greed").
Fortune Lady Engine
"Fortune Lady Water" allows the player to draw two cards when it is special summon to the field, while another face-up Fortune Lady is on the field. This can be easily achieved with Fortune Lady swarming capability. Fortune Lady Water can be special summon by the following cards: Summoner Monk, Call of the Haunted, Monster Reborn, Fortune Lady Dark and Fortune Lady Light. In fact, when combined with Inferno Reckless Summon, it allows the special summoning up to 3 Fortune Lady Water, allowing the player to draw up to 6 cards in total.
Fortune Future allows the player to draw 2 cards by sending a Fortune Lady removed from play to the grave. Fortune Lady can be removed by cards like Future Vision, Interdimensional Matter Transporter, Dimensionhole and Allure of Darkness. Fortune Future combo well with Allure of Darkness and Fortune Lady Dark, draw 2 cards and then remove Fortune Lady Dark from hand, then use Fortune Future to send Fortune Lady dark from removed from play to grave to draw 2 cards.
A great combo involves: using Miracle Fusion or Fusion Gate to remove Elemental Hero or Destiny Hero and Fortune Lady Water to special summon Elemental HERO Absolute Zero. Fortune Future is then used to send the removed Fortune Lady Water to draw 2 cards. Presumably another Fortune Lady is on the field, use reviving cards like Call of the Haunted to special summon Fortune Lady Water to draw 2 more cards or 6 if combined with Inferno Reckless Summoning. Fortune Lady Water can then be tribute for Fortune Lady Dark. When Fortune Lady Dark destroys a monster, Fortune Lady Water can then be special summoning, allowing the players to draw 2 cards. This combo gives you a drawing of 6 cards (10 if combined with Inferno Reckless Summoning) and also swarming of strong monsters.
Level 1 Synchro Material Engine
Most commonly used in Synchro-based decks that can abuse both "Tuningware" and "Darksea Rescue", which allow player to draw 1 card for each of them when used for Synchro Summon. Coupled with "Inferno Reckless Summon" to gather materials and "De-Synchro" for repeat the Synchro Summoning process, allowing player to gain massive hand advantage (especially when combined with "Formula Synchron" and "T.G. Hyper Librarian").
Gusto Serenity Engine
This draw engine utilize "Mind Master" for Special Summoning "Caam, Serenity of Gusto", which can be Tributed over and over and using Caam's effect to shuffle the other 2 Caams in your Graveyard and netting a free draw at the same time, combined with "Brain Research Lab" or "Telekinetic Charging Cell" for avoiding Mind Master's cost, resulting in enormous draw engine that even able to draw your entire deck in just one turn.
However, currently "Mind Master" is Forbidden, making this draw engine no longer viable as a whole, but Caam herself can be utilized as additional draw power for any deck that can utilize Gusto searchers.
Fairy Draw Engine
After getting the Tethys, Goddess of Light, use the effect to keep on Drawing multiple Fairy-Type monsters until your last card is something else. Since most Fairy-Type decks have a heavy number of fairies, then this Effect should be able to draw through a good bit of your Deck.
Pot of Dichotomy Engine
With the right Deck build, "Pot of Dichotomy" can become a powerful drawing force. This engine is possible in any deck that runs some off-type hand traps and an off-type Extra Deck, but it truly shines in those that benefit from milling cards to the graveyard. For example, by maxing out on both "Card Trooper" and Dichotomy, you get a way to draw and attack safely with the former while filling your graveyard for the latter. If your deck doesn't primarily run Machine-Type Monsters, Trooper even helps by filling the role of 1 of the 3 Types. This Trooper-Dichotomy duo can be especially useful in EARTH, Psychic, X-Saber, Naturia, and similar decks.
Since "XYZ-Dragon Cannon", "VWXYZ-Dragon Catapult Cannon", "ABC-Dragon Buster", and "AtoZ Dragon Buster Cannon" require cards to be discarded in order to activate their effects, if you control a face-up "Saffira, Queen of Dragons" when you activate their effects (as those of "ABC Dragon Buster" and "Atoz Dragon Buster Cannon" can be used during either player's turn), you will be able to activate any of Saffira's effect during the End Phase of each and every turn, and potentially draw 2 cards each time in the process, or return discarded LIGHT monsters to your hand to discard again the next turn if you do not need to draw cards. See also: Card Tips:ABC-Dragon Buster.