Howdy, and welcome to my Gragas guide.
First a little bit about Gragas, he's a wonderful champion, and if played right, he can be deadly. I saw, that there was a lack of proper Gragas guids on this side, and as I am a huge fan of him, myself, I decided to create my own, and share it whit you guys.
In this guide, I will explain about how I prefer to build Gragas, and hopefully learn you guys a trick or two. This is oriented for solo top lane, mostly, but can also be used in mid. Yes, you did hear right, mid. I recommend you to read it whole, else you won't get anything out off it.

You may ask, why solo lane?
Cuz if you lane whit another, you get less minion kills, and therefore worse end-game. Also, Gragas is great at solo lane, due his Drunken Rage and Happy Hour.
Additionally, if your laning whit a partner, they will often complain about you, getting all the creep kills.


Let's get to it.

Pros + Great farmer early, and late game + Slow, but great harrassing + Long range ability, Barrel Roll + Escape mechanism + Can charge through obstacles with Body Slam + Easy to play, hard to master + Good laner + Can regenerate mana with Drunken Rage, and health whit Happy Hour + Can slow enemies

Cons - Very slow movement - Requires skill to aim his Barrel Roll - Drop mana pools quick

• Ghost Great escape mechanism, and good for chasing. Really helps Gragas early game, when he is slower than any other champions. I prefer Ghost, instead of Flash, because in length Ghost make you get farer away, than Flash would do. Also, Gragas has already a ability that reminds of Flash; his charges. You don't need the blink over wall, cuz you have your Body Slam for that.
• Exhaust Great if your chasing someone, get in range whit a Body Slam, and pop a Exhaust on him. Exhaust is also great if you fight against champions as Master Yi and Tryndamere, or all other basic attack based champions, cuz you lower their attack damage heavily.
• Cleanse If your getting Exhausted, or hit by other Crowd-Control abilities, Cleanse becomes quite handy. It can allow you to break out of the ability cast on you, and is one of the greatest escape mechanism in League of Legends.
• Flash A very useful spell on mostly every champion. Though I would prefer to take Ghost instead, as Flash is mostly used to blink over wall, which Gragas is able to, whit his Body Slam.
• Ignite Many peoples go for it, as Gragas, and it is indeed a good spell. But I would take other abilities forth from that, as you have a pretty long range on your [[Barrel Roll], that can take out fleeing enemies low on health.

I go 9/0/21 when I build Gragas AP, to get some utility and regeneration, and some AP and spell penetration.

• Perseverence So I can spam my barrels, and for better laning. Whit Perseverence and Happy Hour and Drunken Rage, you can stay far longer in lane, than any other champions.
• Haste To increase the speed of my Ghost, so I can chase far better, or have a better chance to flee.
• Expanded Mind Again, so I can spam my barrels alot early game.
• Meditation To regain mana.
• Utility Mastery So I can keep up my Blue Buff longer.
• Quickness To make up, Gragas missing speed. If you buy Boots instead of Sapphire Crystal, you should spend your masteries in Awareness instead of Quickness .
• Intelligence To reduce the cooldown on my Explosive Cast.
• Presence of the Master So I can pop up, summoner spells far more often.

• Cripple Making Exhaust alot more useful, and reduces their magic resistance, so I hit harder whit Barrel Roll] and [[Explosive Cask.
• Archmage's Savvy Again, to make my [Barrel Roll]] hit harder.
• Sorcery So I can throw Explosive Cask more often.
• Archaic Knowledge So I penetrate high magic resistance champion's magic resi, as Nasus for an example, and other tanks.

This is how my runepage looks like.
I take Greater Mark of Magic Penetration, which is flat magic penetration. This gives me 8.5 magic penetration, from the start of match. Every champion starts with 30 magic resistance. Whit this runes, and Sorcerer's Shoes, I (almost) penetrated through their base magic resistance! This makes your Barrel Roll and Explosive Cask hit hard as hell.

I take Greater Seal of Scaling Ability Power, which grants 0.10 ability power per level, which ends up whit 1.8 at level 18. If you got nine of these, you get a maximum of 16,2 ability power. It may not sound as that of much, but I can tell you, it's a noteworthy different.

I take Greater Glyph of Scaling Ability Power, again to increase my ability power per level. It ends up granting 27,54 ability power, for you!

At last, I take Greater Quintessence of Scaling Ability Power that also grants me ability power per level. You may think its a bit of much ability power, why not some mana regeneration, or cooldown reduction, for Explosive Cask. The answer is, that we focus on Barrel Roll in this guide, we build up mana items, and get the ability power from runes in early game.

• Sapphire Crystal I start of whit this, so my Barrel Roll get alot more spamable, and farm whit it, without worry about loosing to much mana. You can buy boots, to handle Gragas slow speed, but then I would suggest you to get MP/5 runes, or flat mana runes, else you'll find yourself running out of mana in no time. I don't suggest you to buy boots, as you got Quickness , which grants 3% movement speed. Whit the Sapphire Crystal, I go for 2 Healing Pots.
  
• Boots This is the next goal. Get them quickly, but not before Sapphire Crystal. In the case you need it, get some more pots. You won't regain that much of health, whit Happy Hour, as you don't have that of many abilities to spam early.
  
• Catalyst the Protector Finish this thing, whit your Sapphire Crystal. As you might have noticed, you haven't got any ability power yet, whit items. But whit your runes, you find yourself hitting hard already, without a single ability power item.
  
• Rod of Ages This is crafted from your Catalyst the Protector, and a Blasting Wand. If you haven't gold to finish it completely, buy Blasting Wand and return to lane, to farm the last needed gold for it.
  Now, you may ask, why I finish this, that of early. Let me explain. Rod of Ages has two passive, one of them based on leveling up, the other increases per minute. Therefore it is great to get it early, as you level pretty quick early-game, and as you want to stack it fully up, early too. I even get Rod of Ages before Sorcerer's Shoes!
  
• Sorcerer's Shoes Finish it from your Boots, now you penetrate through all their base magic resistance, whit your flat magic penetration, and this.
  
• Rylai's Crystal Scepter Build this from Blasting Wand and Giant's Belt. I prefer to get Blasting Wand as first recipe, and then take Amplifying Tome. You won't need the health from Giant's Belt, as your not going to charge into battle, or get hit that much.
  
• Rabadon's Deathcap Is the next big item. Try to lane, until you have 1600 gold. It might be a pain, as your enemies building up items, while you save them up. But at last, it worth it all. Whit Rabadon's Deathcap your Barrel Roll] and [[Explosive Cask will wipe your enemy off. The 30% extra ability power is just amazing, as it also benefits your ability power through runes!
  
• Lich Bane Would be a alternative, or a final item to build. If your game still go on, I would suggest you to buy this, or/and a Void Staff. Lich Bane grants you a quit nice amount of movement speed, and the Void Staff 40% magic penetration.

This is the way I prefer to build him. There is other ways, but I find him most useful when he burst out quick damage.

As said earlier, Gragas is a great farmer. He has alot of Area of Effect damage, which means his abilities hits over a larger area. Because of his many Area of Effect abilities, he can take out a creep wave quite fast, as he can hit multiply units at once, whit one ability.

Now let me tell short how to farm minion waves fast.
Normally I stay in bushes, waiting for a minion or two to get low on health. When they do so, I either charge out and hit them whit my Body Slam, or run out to hit them whit a basic attacks. Beware, don't run out if a enemy is nearby, as you don't want to get in close range that early in game. It requires patience to play AP Gragas. Don't rush, but wait for your enemy to a stupid move, or to pull to far back, so you can get a minion last hit. If your completely pushed, try to last hit whit Barrel Roll. You may find this, the easiest way to do it, but the price is a huge amount of mana. Whenever you got Blue Buff, begin to use your Barrel Roll as a basic ability to farm creeps. Start off by landing a Barrel Roll between the melee minions, and ranged minions, so the AoE hit all targets. Use your charge, to either get the last hit, or bringing them very low on health, so you can finish them whit some quick basic attacks.
If your not have Blue Buff, try to wait using your Barrel Roll to your enemy get in a position, where you can hit them, and their minion wave whit one single roll.
The key to a good late-game, is alot of minion kills, and therefore alot of golds to spend on items. The more gold, the better you end up.

Here I will explain how to play a more defensive kind of Gragas, where I spam Barrel Roll, and get my last hits on minions.

Early-game: Keep harassing the enemy champion(s). Roll a barrel at them, and return. Try to prevent getting caught up in a creep wave, while the enemy champion is nearby. Whit Gragas slow speed, it will take quite of time to get out of it, and it would be spoil of Ghost. Remember, Body Slam don't charge through minions, it will stop while colliding whit one. Keep this on, and only run to creeps to get last hits, or if the enemy champion is pushed alot backwards. Use your Barrel Roll to harass the enemy constantly, and make them return to base. After a while, when you brought the enemy low on health by harassing, try to play a little more aggressive, but not to much. You don't want to be ganked that early. Wait to the enemy does a stupid move, which get them in range for a Explosive Cask] or a [[Body Slam --> Barrel Roll. Patience is the other key to Gragas, as stated earlier.
While laning, remember to keep up Drunken Rage, and you'll find yourself on full health, when the enemy is under half or less. Because of Happy Hour + Drunken Rage, you can pretty much stay in lane forever, if mana is used correctly. When you got your Explosive Cask begin to play alot more aggressive.

Mid-game: Keep spamming your barrels, and grab a blue buff. Ask for help, in the case you need it. A Gragas whit blue buff, is not something you want to lane against, as a enemy. Whit blue buff is on, you should use Barrel Roll constantly, to both last hit minions, and harass.
Map awareness is the next thing I want to talk about. If your pushing the enemy to his/her turret, you should keep track of what's going on, on the map. If necessarily buy a ward, and place it in the river bushes, to prevent getting ganked by their jungler. If you getting caught between a enemy turret, and a enemy champion under a gank, it becomes quite dangerous. Here Body Slam and Ghost come in use. Put up a Ghost, and charge slightly past him, to slow him, but not colliding whit him. Now if he tries to chase you, keep running. The Ghost should be enough speed, to get away from him, whit the extra speed granted whit Haste .

Late-game: In late game, you should be able to deal tons of damage, and therefore you might be targeted first in smaller teamfights (unless your carry is there too). Stay out of range, and spam your barrels. Remember to have Drunken Rage up, all the time. Many times, I find my self being attacked, without having Drunken Rage up, and I'm dead in no time. If you know your about to encounter a teamfight, drink before the fight. If you use Drunken Rage in fight, the enemy have whole 1 second to hit you, without you doing any actions. If your team is about the flee, and the enemy chases them, throw your Explosive Cask infront of the enemy. I repeat, In front. If you end up placing it wrong, it will actually pull the enemy team, too you, and you won't become popular of that move... So be sure when, and where to throw your Explosive Cask, and if your not sure if you can't aim it, don't even try to. You just end up getting your while team killed. After all it shouldn't be that hard to aim.

At last, I while give you some basic tips.

• Never encounter a teamfight, or any other fight, without having Drunken Rage up.
• When you have Blue Buff spam all your abilities, even if there is no enemy nearby. This will regain health, through Happy Hour.
• Play very passively in the beginning of match.
• Never encounter a teamfight, that you know you get killed in (Example 2v5), instead pull back. Don't mind if the other friend blame you for not helping, as long as you didn't get harmed. This doesn't mean, you should just let your teammates die on purpose.
• Use Explosive Cask to help friends in needs, by throwing it, to blow the enemies away. A well placed Explosive Cask can save a single friend, from the whole enemy team.
• Try to time your Drunken Rage so you drink it just after you threw a Barrel Roll.

• Use Barrel Roll to harass enemies at range, and frustrate them. Always pull a little back, after rolling one.
• Use Barrell Roll to farm creeps. If it is placed right, it can hit a whole minion wave. Try to place it between ranged and melee ones.
• Try to place your Barrel Roll behind the enemy. The enemy champion will react of instinct, and will therefore run backwards, as they see the barrel rolling towards them; and therefore they run into where the barrel actually lands. Remember, enemy team can't see where the barrel lands, take advantage of that.
• In team fights, try to place Barrel Roll in the middle, so it can hit more than one champion at time.

• Use Drunken Rage immediately after using Barrel Roll to regenerate some of the mana cost of it.
• Whenever the duration of it runs out, put up Drunken Rage again. You should never run around, without having drunken rage on.
• When pushing tower, use Drunken Rage to get increased basic attack damage, to deal more damage to the tower.

That was all, I got for you guys. Hope you enjoyed, and learned something out of this wall of text. Tell me, if it could be edited, or if you wan't a additionally category, or just have another random question to me. Might gonna upload some game play, or clips where I show you how to farm, later on. Peace out, and vote, vote, vote, if you did like him!
(Don't -vote if haven't tried it yet, else give me a valid reason.)
Cheers. Corekodo :)